aa r X i v : . [ m a t h . R A ] J a n IDEAL RING EXTENSIONS AND TRUSSES
RYSZARD R. ANDRUSZKIEWICZ, TOMASZ BRZEZI ´NSKI, AND BERNARD RYBO LOWICZ
Abstract.
It is shown that there is a close relationship between ideal extensionsof rings and trusses, that is, sets with a semigroup operation distributing over aternary abelian heap operation. Specifically, a truss can be associated to every el-ement of an extension ring that projects down to an idempotent in the extendingring; every weak equivalence of extensions yields an isomorphism of correspondingtrusses. Furthermore, equivalence classes of ideal extensions of rings by integers arein one-to-one correspondence with associated trusses up to isomorphism given by atranslation. Conversely, to any truss T and an element of this truss one can associatea ring and its extension by integers in which T is embedded as a truss. Consequentlyany truss can be understood as arising from an ideal extension by integers. The keyrole is played by interpretation of ideal extensions by integers as extensions defined bydouble homothetisms of Redei [L. Redei, Die Verallgemeinerung der Schreierschen Er-weiterungstheorie, Acta Sci. Math. Szeged , (1952), 252–273] or by self-permutablebimultiplications of Mac Lane [S. Mac Lane, Extensions and obstructions for rings, Illinois J. Math. (1958), 316–345], that is, as integral homothetic extensions . It isshown that integral homothetic extensions of trusses are universal as extensions oftrusses to rings but still enjoy a particular smallness property: they do not containany subrings to which the truss inclusion map corestricts. Minimal extensions oftrusses into rings are defined. The correspondence between homothetic ring exten-sions and trusses is used to classify fully up to isomorphism trusses arising from ringswith zero multiplication and rings with trivial annihilators. Contents
Part 1. Prelude
21. Introduction 22. Preliminaries 5
Part 2. Extensions
63. From ideal extensions of rings to trusses 64. From trusses to ring extensions 16
Part 3. Interpretation
Date : January 26, 2021.2010
Mathematics Subject Classification.
Key words and phrases.
Truss; ring; extension.
6. Minimality of homothetic extensions of trusses 27
Part 4. Classifications
Coda
Part Prelude Introduction
A truss [Br19], [Br20] is an algebraic system consisting of a set with a ternaryoperation making it into an abelian heap [Pr24], [Ba29] and an associative binaryoperation that distributes over the ternary one. From the universal algebra point ofview its composition involves less operations, so it is simpler, than that of a ring (whichconsists of two binary operations, one unary operation and one nullary operation) ora (two-sided) brace [Ru07], [CJO14], [GV17], which as a set with two entangled groupstructures involves six operations and whose connection with the set-theoretic Yang-Baxter equation has led to a remarkable surge in interest recently. Yet a truss equippedwith a specific nullary operation or with an element with special properties can be madeinto a ring; if the binary operation is a group operation, then there is a (two-sided)brace associated to a truss. Conversely, every ring can be made into a truss in a naturalway, by associating the (unique) heap operation [ a, b, c ] = a − b + c to the abelian groupoperation, and so can every brace. Thus trusses are both simpler in architecture andmore general than rings. Alas their definition involves a ternary operation that isless intuitive and familiar than binary operations, and so one is faced with a familiardilemma of generality versus comprehension.In mathematics as in any other human endeavour there is a natural tendency tofamiliarise what is new or unknown by contrasting or comparing it with what is well-known. Thus the desire to see how trusses are related to rings or how trusses canbe described in ring-theoretic terms is most understandable. The key observation ofRump [Ru07] that a two-sided brace can be made into a radical ring by modifying oneof its operations (and vice versa, a radical ring gives rise to a two-sided brace) hasbeen extended to trusses in [Br19], [Br20] with a caveat that a central element mustexist; the resulting ring is not necessarily a radical ring. In this paper we show that aring can be associated to any truss and any element, but we go further than that. Weshow that a natural framework for ring-theoretic studies of trusses is provided by idealring extensions (see, for example, [Pe85]), in particular those that arise from Redei’s DEAL RING EXTENSIONS AND TRUSSES 3 homothetisms [Re52] or Mac Lane’s self-permutable bimultiplications [Ma58]. In short,we show that a truss can be associated to every element of an extension ring thatprojects down to an idempotent in the extending ring, while every weak equivalence ofextensions yields an isomorphism of corresponding trusses. Furthermore, equivalenceclasses of extensions of rings by integers are in one-to-one correspondence with the classof associated trusses up to isomorphism given by translation. Conversely, to any truss T and an element of this truss one can associate a ring together with its extension byintegers in which T is embedded as a truss (and also as a paragon or an equivalence classof a congruence of rings). Since every ideal extension of a ring by integers is equivalentto an extension by Z through a double homothetism, any truss can be understood asarising from such a homothetic extension, i.e. every truss is a homothetic truss . Notein passing that all (not only those by integers) ideal ring extensions arise from familiesof permutable bimultiplications or amenable homothetisms by the Everett theorem[Ev42] (see [Pe85] for an elegant presentation and simplification of the proof).The paper is divided into four parts and a short coda. The first part (which includesthis introduction) contains preliminary definitions, in particular basic notions from thetheory of heaps and trusses. The second part gathers main results of the paper. We be-gin by recalling the definitions of ring extensions and their equivalences. An extension( ϕ R , S, ϕ Z ) consists of a ring monomorphism ϕ R : R −→ S and a ring epimorphism ϕ Z : S −→ Z such that ker ϕ Z = Im ϕ R (unless stated otherwise, by a ring we meanan associative ring not necessarily with identity). Following the terminology of MacLane [Ma58] we refer to such an extension as to an extension of R to S by Z (notethat an opposite terminology is often used in homological algebra). Two extensions( ϕ R , S, ϕ Z ) and ( ϕ ′ R , S ′ , ϕ ′ Z ) are equivalent if there is an isomorphism between S and S ′ that commutes with the identity automorphisms on R and Z . Furthermore, we saythat ( ϕ R , S, ϕ Z ) and ( ϕ ′ R , S ′ , ϕ ′ Z ) are weakly equivalent, provided that they are equiva-lent up to an automorphism of R . We observe in Proposition 3.2 that if there is q ∈ S such that q − q ∈ ϕ R ( R ), then q + ϕ R ( R ) is a sub-truss of the truss associated to S . Furthermore, any weak equivalence map for extensions restricts to an isomorphismof corresponding trusses. We then proceed to focus on a class of extensions arisingfrom double homothetisms or self-permutable bimultiplications, i.e. pairs of additiveendomorphisms σ = ( → σ , ← σ ) of a ring R satisfying a number of associativity-like con-ditions; see Definition 3.3. In Theorem 3.6 we associate an extension of a ring R byintegers (or by integers modulo the exponent of the abelian group of R if finite) to apair ( σ, s ) consisting of a double homothetism σ on R and an element s ∈ R such that σs = sσ and σ − σ is the bimultiplication by s . We term the resulting extension of R , denoted by R ( σ, s ), a homothetic extension of R (infinite in the integral case andcyclic or finite in the modular case). Next we note that in fact every ring extensionby Z is equivalent to an infinite homothetic extension and that every R ( σ, s ) induces atruss T ( σ, s ) on the heap corresponding to the abelian group of R . We term T ( σ, s ) a homothetic truss . In a couple of lemmas leading to Theorem 3.11 we describe isomor-phisms of homothetic trusses and connect them with (weak) equivalences of homotheticextensions: Two extensions are weakly equivalent if and only if corresponding trussesare isomorphic, while equivalences of extensions correspond to isomorphisms of trussesgiven by translations by an element. RYSZARD R. ANDRUSZKIEWICZ, TOMASZ BRZEZI ´NSKI, AND BERNARD RYBO LOWICZ
In Section 4 we take the opposite route: from trusses to ring extension. The firstmain result of this section is Theorem 4.3, which allows one to associate a ring to anytruss T and an element e ∈ T . We denote this truss by R( T ; e ). This ring has additionobtained by retracting of the ternary operation in T at e (so it has an abelian groupstructure derived from the heap structure of T ), but with a modified multiplication.There are no assumptions on e , and thus Theorem 4.3 provides one with a generalisationof the construction presented in [Br19], which associates a ring to a central element ofa truss or the construction of Rump [Ru07] connecting two-sided braces with radicalrings. The way to recover the original multiplication of T is described in Theorem 4.8: e determines a homothetic datum ( ε, e ) on R( T ; e ), and T is the truss embedded inthe corresponding homothetic ring extension of R( T ; e ), denoted by T ( e ) in the infinitecase or T c ( e ) in the cyclic case, that is, T = T( ε, e ). The results of Part 2 thus can besummarised as one of two main messages of this paper: every truss is a homothetictruss; every ring is a ring associated to a truss with a fixed element. Part 3 on one hand gives a categorical interpretation of the infinite homothetic ringextensions T ( e ) in which a truss T embeds, while on the other attempts at charac-terisation of smallest rings into which T embeds. In Theorem 5.2 we construct a ringisomorphism between T ( e ) and the universal ring T obtained from a truss T by adjoin-ing the zero element [BR20a, Lemma 3.13]. As a consequence, the infinite homotheticextension T ( e ) has the following universal property: any truss homomorphism from T to a ring R factorises through the inclusion T ֒ → T ( e ) and a unique ring homomorphism T ( e ) −→ R , i.e. there exists a universal arrow from T to the functor T : Ring −→ Trs (see [Ma98, Section III.1]). Section 6 outlines various possible notions of ‘smallness’of a ring that contains T . The most crude one is proposed in Definition 6.1: a locallysmall extension of T is given by a ring which has no proper subrings containing T (asa sub-truss). The infinite homothetic extension of T into ring T ( e ) has this property.The intermediate notion given in Definition 6.7 distinguishes those small extensions inwhich T is isomorphic (as a heap) with an essential ideal. Finally, a minimal extensioncorresponds to ideals in T ( e ) which intersect trivially with the ideal induced by thecanonical inclusion of T into T ( e ). All these notions and differences between them areillustrated by examples.Part 4 uses results of Part 2, in particular that every truss is a homothetic truss, toclassify all trusses corresponding to rings with zero multiplication (Section 7) and zeroannihilators (Section 8). In the first case, in which all the structural information isnecessarily contained in the abelian group of the ring, we show that there is a one-to-one correspondence between isomorphism classes of trusses corresponding to rings withzero multiplication on an abelian group A and ordered direct sum decompositions of A into four subgroups; Theorem 7.5. In the latter case there is a bijective correspondencebetween isomorphism classes of homothetic trusses on R and equivalence classes ofidempotents in the ring Ξ( R ) of outer bimultiplications on R (with respect to therelation defined in Definition 3.5); see Theorem 8.2. In particular, and quite surprising,there are exactly two isomorphism classes of trusses on one-sided maximal ideals insimple rings with identity; see Theorem 8.3. This seems to be a rather unexpectedapplication of the theory of maximal essential extensions developed by Beidar [Be85],[Be93]. All the results of this part are employed to give a full classification of non-isomorphic trusses with the heap structure corresponding to the abelian group Z p × Z p . DEAL RING EXTENSIONS AND TRUSSES 5 Preliminaries
We start by gathering in one place key information about heaps and trusses, and byestablishing the notation. Further details can be found in e.g. [Br20].An abelian heap is a set H with a ternary operation [ − , − , − ] such that, for all h i , i = 1 , . . . ,
5. [ h , h , [ h , h , h ]] = [[ h , h , h ] , h , h ] , (2.1a)[ h , h , h ] = h & [ h , h , h ] = h , (2.1b)[ h , h , h ] = [ h , h , h ] . (2.1c)Equation (2.1a) expresses the associative law for heaps, equations (2.1b) are known asMal’cev identities, and equation (2.1c) is the heap commutative law. In view of theseaxioms the distribution of brackets in multiple applications of the ternary operationdoes not play any role, and hence we write[ h , h , . . . , h n +1 ]for the element of H obtained by any possible application of the ternary operationto the (always odd) 2 n + 1-tuple ( h , h , . . . , h n +1 ) ∈ H n +1 . Furthermore, equa-tions (2.1b) and (2.1c) yield the following cancellation and rearrangement rules, for all h , h , . . . , h n +1 ∈ H ,[ h , . . . , h i − , h i , h i , h i +1 , . . . h n ] = [ h , . . . , h i − , h i +1 , . . . h n ] , (2.2a)[ h , h , . . . , h n +1 ] = [ h ̟ (1) , h ς (2) , h ̟ (3) , . . . , h ς (2 n ) , h ̟ (2 n +1) ] , (2.2b)for any permutation ̟ on the set { , , , . . . , n + 1 } and any permutation ς on { , , . . . , n } .For any e ∈ H , the set H with the binary operation + e = [ − , e, − ] is an abeliangroup, known as a retract of H . The chosen element e is the zero for this group andthe inverse − e h of h is [ e, h, e ]. We denote this unique up to isomorphism group byG( H ; e ). Conversely, for any (abelian) group G , the operation [ a, b, c ] = a − b + c definesthe heap structure on G ; we denote this heap by H( G ). A homomorphism of heaps isa function that preserves heap operations. In particular, any group homomorphism isa heap homomorphism for the corresponding heaps so the assignment H : G H( G )is a functor from the category of (abelian) groups to the category of (abelian) heaps.A truss is a set T together with a ternary operation [ − , − , − ] and a binary oper-ation · (denoted by a juxtaposition of elements and called multiplication) such that( T, [ − , − , − ]) is an abelian heap, ( T, · ) is a semigroup and · distributes over [ − , − , − ],that is, for all a, b, c, d ∈ T , a [ b, c, d ] = [ ab, ac, ad ] & [ a, b, c ] d = [ ad, bd, cd ] . (2.3)A morphism of trusses is a function that is a homomorphism of both heaps and semi-groups. Unless stated otherwise, by a ring we mean an associative ring not necessarilywith identity. To any ring ( R, + , · ) one can associate a truss T( R ) with the heapstructure H( R, +) and the original multiplication. The assignment T : R T( R )and identity on morphisms is a functor from the category of rings to the categoryof trusses. Conversely, if a truss T has an absorber , that is, an element e such that ea = ae = e , for all a ∈ T , then R( T ; e ) := (G( R ; e ) , · ) is a ring and T = T(R( T ; e )). RYSZARD R. ANDRUSZKIEWICZ, TOMASZ BRZEZI ´NSKI, AND BERNARD RYBO LOWICZ
An equivalence class of a congruence in a truss T is called a paragon . Equivalently,a paragon P is a sub-heap of ( T, [ − , − , − ]) such that, for all p, q ∈ P and a ∈ T ,[ ap, aq, q ] ∈ P & [ pa, qa, q ] ∈ P. (2.4)A sub-heap satisfying the first of equations of (2.4) is called a left paragon and onethat satisfies the second of these equations is called a right paragon . Part Extensions From ideal extensions of rings to trusses
This section contains the first main results of the paper. We begin by recalling thenotions of ideal extensions of rings and (weak) equivalences between such extensions.Next we show that one can associate a truss to every element of an extension ring thatyields an idempotent in the ring by which the extension is achieved. Weak equivalencesof extensions restrict to isomorphisms of these trusses. In the converse direction weconstruct (infinite and finite) extensions of a given ring by double homothetisms, andshow that such extensions are weakly equivalent if and only if the associated trussesare isomorphic and equivalent if and only if the associated trusses are isomorphic by atranslation or translationally isomorphic . We also observe that any infinite homotheticextension is equivalent to an extension by the ring of integers.The following definition is taken from [Ma58] (see also [Pe85]).
Definition 3.1.
An exact sequence of ring homomorphisms0 / / R ϕ R / / S ϕ Z / / Z / / , is called an ideal ring extension of R to S by Z or simply a ring extension . We write( ϕ R , S, ϕ Z ) for this ring extension.Two extensions ( ϕ R , S, ϕ Z ) and ( ϕ ′ R , S ′ , ϕ ′ Z ) are said to be equivalent if there existsa ring isomorphism Θ : S −→ S ′ rendering the following diagram commutative R ϕ R / / ϕ ′ R (cid:15) (cid:15) S ϕ Z (cid:15) (cid:15) Θ x x ♣♣♣♣♣♣♣♣♣♣♣♣♣ S ′ ϕ ′ Z / / Z. (3.1)We write ( ϕ R , S, ϕ Z ) Θ ≡ ( ϕ ′ R , S ′ , ϕ ′ Z ).Two extensions ( ϕ R , S, ϕ Z ) and ( ϕ ′ R , S ′ , ϕ ′ Z ) are said to be weakly equivalent if thereexist a ring isomorphism Θ : S −→ S ′ and a ring automorphism Θ R : R −→ R suchthat ( ϕ R , S, ϕ Z ) Θ ≡ ( ϕ ′ R ◦ Θ R , S ′ , ϕ ′ Z ). In that case we write ( ϕ R , S, ϕ Z ) Θ ∼ = ( ϕ ′ R , S ′ , ϕ ′ Z )The reader should be made aware that particularly in the texts in homological algebra(see e.g. [We94, Section 9.3]) the sequence in Definition 3.1 is referred to as an extensionof Z to S by R . We have elected here to chose the conventions of Mac Lane [Ma58]. Proposition 3.2.
Let ( ϕ R , S, ϕ Z ) be an ideal ring extension and let q ∈ S . Then DEAL RING EXTENSIONS AND TRUSSES 7 (1) The set q + ϕ R ( R ) is a sub-heap of H( S ) and a paragon of T( S ) .(2) The sub-heap q + ϕ R ( R ) is a sub-truss of T( S ) if and only if q − q ∈ ϕ R ( R ) (3.2) This truss is denoted by T ( ϕ R , S, ϕ Z ; q ) .(3) If ( ϕ R , S, ϕ Z ) Θ ∼ = ( ϕ ′ R , S, ϕ ′ Z ) , then the map Θ restricts to the isomorphism oftrusses T ( ϕ R , S, ϕ Z ; q ) ∼ = T ( ϕ ′ R , S ′ , ϕ ′ Z ; Θ( q )) .Proof. Since ϕ R ( R ) = ker ϕ Z is an ideal in S , for all q ∈ S , q + ϕ R ( R ) is an equivalenceclass of a congruence relation in S , hence it is a paragon in T( S ) by [BR20b, Corol-lary 3.3]. Condition (3.2) is equivalent to the statement that ϕ Z ( q ) = q + ϕ R ( R ) is anidempotent in Z , hence ( q + ϕ R ( R )) = q + ϕ R ( R ) is closed under the multiplicationin S .Since Θ is a ring isomorphism, its restriction to T ( ϕ R , S, ϕ Z ; q ) is a monomorphismof trusses. We need to show that Θ( T ( ϕ R , S, ϕ Z ; q )) = T ( ϕ ′ R , S ′ , ϕ ′ Z ; Θ( q )). For all r ∈ R ,Θ( q + ϕ R ( r )) = Θ( q ) + Θ( ϕ R ( r )) = Θ( q ) + ϕ ′ R (Θ R ( r )) ∈ T ( ϕ ′ R , S ′ , ϕ ′ Z ; Θ( q )) . The commutativity of the diagram (3.1) and the fact that Θ R is an automorphism ofrings yield the surjectivity of Θ. This completes the proof of the proposition. (cid:3) Since weakly equivalent extensions give rise to isomorphic trusses, Proposition 3.2provides one with a method of constructing (isomorphism classes of) trusses. In fact onecan look for a statement in the opposite direction, that is, for types of extensions whose(weak) equivalence classes are determined by (isomorphism classes of) correspondingtrusses. To this end we need to focus on extensions of a more specific kind.
Definition 3.3.
Let R be a ring and let σ be a double operator on R , that is a pairof additive endomorphisms, → σ : R −→ R, a σa, ← σ : R −→ R, a aσ. (1) The double operator σ is called a bimultiplication [Ma58] or a bitranslation [Pe85]if, for all a, b ∈ R , σ ( ab ) = ( σa ) b & ( ab ) σ = a ( bσ ) , (3.3a) a ( σb ) = ( aσ ) b. (3.3b)The set of all bimultiplications is denoted by Ω( R ).(2) A bimultiplication σ is called a double homothetism [Re52] or is said to be self-permutable [Ma58] provided that, for all a ∈ R ,( σa ) σ = σ ( aσ ) . (3.4)The set of all double homothetisms on R is denoted by Π( R ).In short, conditions (3.3a) mean that → σ is a right and ← σ is a left R -module homo-morphism. A bimultiplication is called simply a multiplication in [Ho47]. In functionalanalysis, in particular in the context of C ∗ -algebras, bimultiplications are known as multipliers [He56], [Bu68]. The relations (3.4) mean that → σ commutes with ← σ in the RYSZARD R. ANDRUSZKIEWICZ, TOMASZ BRZEZI ´NSKI, AND BERNARD RYBO LOWICZ endomorphism ring End( R, +). The set Ω( R ) is a unital ring with the addition andmultiplication, for all σ, σ ′ ∈ Ω( R ), a ∈ R ,( σ + σ ′ ) a = σa + σ ′ a, a ( σ + σ ′ ) = aσ + aσ ′ , (3.5a)( σσ ′ ) a = σ ( σ ′ a ) , a ( σσ ′ ) = ( aσ ) σ ′ . (3.5b)In particular in the context of C ∗ -algebras, Ω( R ) is known as a multiplier algebra . Notethat the rules (3.5b) mean the composition of right-linear components of bimultiplic-tions and opposite composition of the left-linear ones. It is clear from (3.5) that R isan Ω( R )-bimodule.In general, Π( R ) need not be a subring of Ω( R ). The rules of Definition 3.3 mean thatwe need not write any brackets in-between letters of the words composed of elementsof R and a homothetism on R .For any a ∈ R , the left and right multiplications by a form a double homothetism,which we denote by ¯ a . That is, for all b ∈ R ,¯ ab = ab, b ¯ a = ba. (3.6)Such a double homothetism is said to be inner and the abelian group of all innerhomothetisms is denote by ¯ R . The right R -module and left R -module components of¯ a are denoted by → a and ← a respectively. Note that if σ is a double homothetism, then,for all a ∈ R , σ + ¯ a is also a double homothetism by equations (3.3). By the sametoken, for all a ∈ R and σ ∈ Ω( R ),¯ aσ = aσ & σ ¯ a = σa. (3.7)This implies that ¯ R is an essential ideal in Ω( R ). The quotient ring Ω( R ) / ¯ R is called thering of outer bimultiplications and is denoted by Ξ( R ) (in context of C ∗ -algebras, Ξ( R )might be referred to as a corona algebra ). The canonical surjection Ω( R ) −→ Ξ( R ) isdenoted by ξ .The following lemma can be proven by direct calculations. Lemma 3.4.
Let Φ be an automorphism of a ring R . For any bimultiplication σ ∈ Ω( R ) define the double operator Φ ∗ ( σ ) on R by −→ Φ ∗ ( σ ): a Φ( σ Φ − ( a )) , ←− Φ ∗ ( σ ): a Φ(Φ − ( a ) σ ) . (3.8) The assignment σ Φ ∗ ( σ ) defines a ring automorphism Φ ∗ on Ω( R ) . Furthermore, Φ ∗ (Π( R )) = Π( R ) , Φ ∗ maps inner homothetisms to inner ones, and, for all a ∈ R , Φ( a ) = Φ ∗ (¯ a ) . (3.9)Equation (3.9) implies that Φ ∗ : Ω( R ) −→ Ω( R ) descends to the ring automorphismΦ ⋄ : Ξ( R ) −→ Ξ( R ) by the diagramΩ( R ) Φ ∗ / / ξ (cid:15) (cid:15) Ω( R ) ξ (cid:15) (cid:15) Ξ( R ) Φ ⋄ / / Ξ( R ) , (3.10)where ξ is the canonical surjection. The following notion will prove particularly helpfulfor discussing trusses associated to rings with trivial annihilators. DEAL RING EXTENSIONS AND TRUSSES 9
Definition 3.5.
Two outer bimultiplications σ, σ ′ ∈ Ξ( R ) are said to be equivalent ifthere exists a ring automorphism Φ : R −→ R such that σ ′ = Φ ⋄ ( σ ) . In that case wewrite σ ∼ σ ′ .With all these notions and notation at hand we are now ready to consider extensionsof our particular interest. Theorem 3.6.
Let R be a ring, and σ ∈ Π( R ) and s ∈ R such that, σs = sσ, (3.11a) σ = σ + ¯ s. (3.11b) Then(1) (a) The abelian group R × Z together with the product, for all a, b ∈ R , k, l ∈ Z , ( a, k )( b, l ) = ( ab + laσ + kσb + kls, kl ) , (3.12) is an associative ring. This ring is denoted by R ( σ, s ) .(b) The sequence / / R ϕ R / / R ( σ, s ) ϕ Z / / Z / / , (3.13) where ϕ R : a ( a, and ϕ Z : ( a, k ) k is an exact sequence of rings.(c) Any ideal ring extension / / R ψ R / / S ψ Z / / Z / / . (3.14) is equivalent to an extension of type (3.13) .(2) If the abelian group ( R, +) has a finite exponent N , then:(a) The abelian subgroup I N = { } × N Z of R × Z is an ideal in R ( σ, s ) . Thequotient ring R ( σ, s ) /I N is denoted by R c ( σ, s ) .(b) The sequence / / R ϕ cR / / R c ( σ, s ) ϕ c Z N / / Z N / / , (3.15) where ϕ cR : a ( a,
0) + I N and ϕ c Z N : ( a, k ) + I N k mod N is an exactsequence of rings.(3) The set { ( a, | a ∈ R } ⊆ R ( σ, s ) (resp. { ( a,
1) + I N , | a ∈ R } ⊆ R c ( σ, s ) in thefinite exponent case) is a sub-truss and a paragon of T( R ( σ, s )) (resp. T( R c ( σ, s )) in the finite exponent case).Proof. That multiplication (3.12) makes R × Z into an associative ring can be checkedby direct calculations that use the double homothetism rules in Definition 3.3 and equations (3.11). Explicitly, for all a, b, c ∈ R and k, l, m ∈ Z ,(( a, k )( b, l ))( c, m ) = ( ab + laσ + kσb + kls, kl ) ( c, m )= ( abc + laσc + kσbc + klsc + mabσ + lmaσ + kmσbσ + klmsσ + klσc + klms, klm )= ( abc + laσc + kσbc + klsc + mabσ + lmaσ + lmas + kmσbσ + klmσs + klσc + klms, klm )= ( a, k ) ( bc + mbσ + lσc + lms, lm ) = ( a, k )(( b, l )( c, m )) . Hence the multiplication is associative. The distributive laws follow from the additivityof σ and distributive laws in R and Z . The statement (1b) follows immediately fromthe definition of R ( σ, s ).To prove (1c) first note that the sequence (3.14) splits as a sequence of abeliangroups, and hence there is the following diagram with exact rows,0 / / R ψ R / / S ζ o o ψ Z / / ¯ π E E π (cid:8) (cid:8) Z κ o o / / , (3.16)in which κ, ζ , π and ¯ π are additive maps such that ψ Z ◦ κ = id Z , π = κ ◦ ψ Z , ¯ π = id S − π = ψ R ◦ ζ , ζ ◦ ψ R = id R . For any q ∈ ϕ Z − (1), define a double operator σ and s ∈ R , by σa = ζ ( qψ R ( a )) , aσ = ζ ( ψ R ( a ) q ) , s = ζ ( q − q ) , (3.17)for all a ∈ R . Observe that, for all a ∈ R , π ( q − q ) = π ( qψ R ( a )) = π ( ψ R ( a ) q ) = 0 , by the exactness of the sequence (3.14). Hence equations (3.17) can be equivalentlywritten as ψ R ( σa ) = qψ R ( a ) , ψ R ( aσ ) = ψ R ( a ) q, ψ R ( s ) = q − q. (3.18)Using (3.18) and the fact that ψ R is an injective ring homomorphism, one easily findsthat conditions (3.11) are satisfied for σ and s defined by (3.17), and so there is anexact sequence such as (3.13).Again a simple calculation aided by (3.18) and the ring monomorphism property of ϕ R yields that Θ : R ( σ, s ) −→ S, ( a, n ) ψ R ( a ) + nq, is an isomorphism of rings with the inverse Θ − ( x ) = ( ζ ( x ) , ψ Z ( x )), which provides onewith the required equivalence of extensions.If ( R, +) has a finite exponent N , then, for all a ∈ R and k, l ∈ Z ,( a, k )(0 , N l ) = ( N laσ + klN s, klN ) = (0 , klN ) ∈ I N , and similarly for the right multiplication. Hence I N is an ideal in R ( σ, s ). DEAL RING EXTENSIONS AND TRUSSES 11
In view of the definition of I N , the map ϕ cR is injective. The kernel of ϕ c Z N consistsof all elements of the form ( a,
0) + I N , i.e. of the whole of the image of ϕ cR . The map ϕ c Z N is obviously surjective. This proves the exactness of the sequence (3.15).Finally, since (0 , = (0 ,
1) + ( s,
0) in both cases (modulo I N in the finite exponentcase), these extensions satisfy assumptions of Proposition 3.2 with q = (0 ,
1) (or q =(0 ,
1) + I N in the cyclic case). The stated subsets are of the form q + ϕ R ( R ) and hencethey are trusses and paragons, as claimed. This completes the proof of the theorem. (cid:3) Definition 3.7.
Let R be a ring.(1) A pair ( σ, s ), where σ ∈ Π( R ) and s ∈ R satisfying conditions (3.11) is called a homothetic datum on R .(2) Let ( σ, s ) be a homothetic datum on R .(i) The extension ( ϕ R , R ( σ, s ) , ϕ Z ) given by the sequence (3.13) is called an integral or infinite homothetic extension .(ii) The extension ( ϕ cR , R c ( σ, s ) , ϕ c Z N ) given by the sequence (3.15) is called a cyclic or finite homothetic extension .(3) The image under the canonical (projection) isomorphism H( R ) × { } −→ H( R )of the truss considered in assertion (3) of Theorem 3.6 is called a homothetic truss on R and is denoted by T( σ, s ).Note that T( σ, s ) is isomorphic to R as a heap and it is a truss since 1 is an idempotentin Z (or Z N ). Explicitly, the multiplication ⋄ in T( σ, s ) is given by a ⋄ b = ab + aσ + σb + s, for all a, b ∈ R . In particular, for the trivial homothetic datum (0 ,
0) on R , T(0 ,
0) =T( R ), the truss associated to the ring R . Next, we identify isomorphism classes ofhomothetic trusses T( σ, s ). Lemma 3.8.
For all ring automorphisms Φ of R and all v ∈ R , T( σ, s ) ∼ = T( σ ′ , s ′ ) ,where s ′ = Φ( s + v + v − vσ − σv ) & σ ′ = Φ ∗ ( σ − ¯ v ) , (3.19) where Φ ∗ is the induced bijection on Π( R ) defined in Lemma 3.4.Proof. First we need to show that the pair ( σ ′ , s ′ ) is a homothetic datum on R . Usingthe fact that ( σ, s ) is a homothetic datum, we can compute,( s + v + v − vσ − σv )( σ − ¯ v )= sσ + v σ + vσ − vσ − σvσ − sv − v − v − vσv + σv = σs + v σ − vs − σvσ − sv − v − v − vσv + σv = σs + v σ − σvσ − vs − σ v + σv − v − v − vσv + σv = ( σ − ¯ v )( s + v + v − vσ − σv ) . Therefore, s ′ σ ′ = Φ( s + v + v − vσ − σv )Φ ∗ ( σ − ¯ v )= Φ (cid:0) ( s + v + v − vσ − σv )( σ − ¯ v ) (cid:1) = Φ (cid:0) ( σ − ¯ v )( s + v + v − vσ − σv ) (cid:1) = Φ ∗ ( σ − ¯ v )Φ( s + v + v − vσ − σv ) = σ ′ s ′ . Hence the condition (3.11a) holds for s ′ and σ ′ . Next, using the fact that Φ ∗ preservesboth addition and multiplication, property (3.9) and that σ and s satisfy (3.11b) wecompute, σ ′ = Φ ∗ ( σ − ¯ v ) = Φ ∗ (( σ − ¯ v )( σ − ¯ v ))= Φ ∗ (cid:0) σ + ¯ s − ¯ vσ − σ ¯ v + ¯ v (cid:1) = Φ ∗ ( σ − ¯ v ) + Φ ∗ (cid:16) s − vσ − σv + v + v (cid:17) = σ ′ + Φ( s − vσ − σv + v + v ) = σ ′ + s ′ , as required. Therefore, there is a homothetic extension R ( σ ′ , s ′ ) and the correspondingtruss T( σ ′ , s ′ ).Consider the map Φ v : R −→ R, a Φ( a + v ) . Since Φ is a ring automorphism Φ v is an automorphism of the heap H( R ). Furthermore,for all a, b ∈ R ,Φ v ( a )Φ v ( b ) + Φ v ( a ) σ ′ + σ ′ Φ v ( b ) + s ′ = Φ( a + v )Φ( b + v ) + Φ (( a + v )( σ − ¯ v ))+ Φ (( σ − ¯ v )( b + v )) + Φ( s − vσ − σv + v + v )= Φ( ab + aσ + σb + s + v ) = Φ v ( ab + aσ + σb + s ) . Therefore, Φ v transforms multiplication in T( σ, s ) into multiplication in T( σ ′ , s ′ ), andhence is the required truss isomorphism T( σ, s ) ∼ = T( σ ′ , s ′ ). (cid:3) Note in passing that the second of conditions (3.19) is equivalent to the statementthat the outer bitranslations ξ ( σ ) , ξ ( σ ′ ) ∈ Ξ( R ) are equivalent in the sense of Defini-tion 3.5. Lemma 3.9.
Let R ( σ, s ) and R ( σ ′ , s ′ ) be homothetic extensions of R such that T( σ, s ) ∼ =T( σ ′ , s ′ ) . Then there exists a ring automorphism Φ of R and an element v ∈ R suchthat the relations (3.19) hold.Proof. Let Ψ : T( σ, s ) −→ T( σ ′ , s ′ ) be a truss isomorphism. DefineΦ : R −→ R, a Ψ( a ) − Ψ(0) . Since Ψ is a heap homomorphism, Φ is an additive map, as, for all a, b ∈ R ,Φ( a + b ) = Ψ( a − b ) − Ψ(0) = Ψ( a ) − Ψ(0) + Ψ( b ) − Ψ(0) = Φ( a ) + Φ( b ) . Clearly, Φ is an automorphism of abelian groups with the inverse Φ − ( a ) = Ψ − ( a ) − Ψ − (0). Set v = − Ψ − (0) ∈ R. DEAL RING EXTENSIONS AND TRUSSES 13
Note that, since Ψ respects the ternary operation, for all a ∈ R ,Φ( a + v ) = Ψ( a − Ψ − (0)+0) − Ψ(0) = Ψ( a ) − Ψ(Ψ − (0))+Ψ(0) − Ψ(0) = Ψ( a ) . (3.20)Since e Ψ is an isomorphism of trusses, for all a, b ∈ R ,Ψ − ( a )Ψ − ( b ) + Ψ − ( a ) σ + σ Ψ − ( b ) + s = Ψ − ( ab + aσ ′ + σ ′ b + s ′ ) . (3.21)Evaluating the equality (3.21) at a = b = 0 we obtain,Ψ − ( s ′ ) = s + Ψ − (0) σ + σ Ψ − (0) + Ψ − (0)Ψ − (0)That is, s ′ = Ψ( s − vσ − σv + v ) = Φ( s − vσ − σv + v + v ) , (3.22)where the last equality follows by (3.20). Therefore, the first of conditions (3.19)holds. Next, setting a = 0 in (3.21) and using (3.22) as well as the fact that Ψ is ahomomorphism of heaps, equation (3.20) and the definitions of Φ, Φ − and v we find σ ′ b = Ψ (cid:0) σ Ψ − ( b ) + σv − v − v − v Ψ − ( b ) (cid:1) = Φ (cid:0) σ Ψ − ( b ) + σv − v − v Ψ − ( b ) (cid:1) = Φ (cid:0) σ Φ − ( b ) − v Φ − ( b ) (cid:1) = Φ ∗ ( σ − ¯ v ) b. In a similar way by setting b = 0 in (3.21) one finds that aσ ′ = a Φ ∗ ( σ − ¯ v ). Puttogether these prove that σ ′ = Φ ∗ ( σ − ¯ v ) , as required. The proof that Φ respects multiplication in R follows the same lines asthe proof that Φ v is a truss homomorphism in Lemma 3.8 (cid:3) Our next task is to connect the correspondences between ring and truss isomorphismsdescribed in Lemma 3.8 and Lemma 3.9 with equivalences of homothetic extensions.Before we do this, however, we would like to make the following observation. Anyendomorphism of abelian groups has at least one fixed point, so, in particular, anytranslation by an element that is not identity, i.e. the function a a + e , for afixed e = 0, is not a group endomorphism. Consequently, there are no translationring endomorphisms other than the identity. In contrast, given a heap H , for any twoelements e, f ∈ H , the translation map τ fe : H −→ H, a [ a, e, f ] , (3.23)is a heap automorphism. This leads us to the following definition. Definition 3.10.
Let T and T ′ be trusses on the same heap. We say that T and T ′ are translationally isomorphic if there exist elements e, f such that the translation heapautomorphism τ fe is an isomorphism of trusses. In that case we write T tr ∼ = T ′ .Translationally isomorphic trusses turn out to play a key role in the study of equiv-alence classes of homothetic ring extensions. Theorem 3.11.
For any ring R ,(1) Two homothetic extensions of R are weakly equivalent if and only if the corre-sponding trusses are isomorphic. (2) Two homothetic extensions of R are equivalent if and only if the correspondingtrusses differ by a translation.Proof. For statement (1) we observe that if homothetic extensions ( ϕ R , R ( σ, s ) , ϕ Z ) and( ϕ ′ R , R ( σ ′ , s ′ ) , ϕ ′ Z ) are weakly equivalent by Θ and Θ R , then :Θ ◦ ϕ R = ϕ ′ R ◦ Θ R implies Θ( a,
0) = (Θ R ( a ) , ,ϕ Z = ϕ ′ Z ◦ Θ implies Θ(0 ,
1) = ( e, , for all a ∈ R and some e ∈ R . HenceΘ( a,
1) = Θ( a,
0) + Θ(0 ,
1) = (Θ R ( a ) + e, , i.e., the ring isomorphism Θ induces an isomorphism of trussesΘ T : T ( σ, s ) −→ T ( σ ′ , s ′ ) , a Θ R ( a ) + e. In the opposite direction, observe that Lemma 3.8 and Lemma 3.9 establish a bijec-tive correspondence between isomorphisms of trusses T( σ, s ) and systems (Φ , v ). Letus assume that T( σ, s ) ∼ = T( σ ′ , s ′ ) and let Φ be the corresponding automorphism of R and v the corresponding element of R . DefineΘ : R ( σ, s ) −→ R ( σ ′ , s ′ ) , ( a, k ) (Φ( a ) + k Φ( v ) , k ) . Clearly, Θ is an isomorphism of abelian groups, so we need to check whether it preservesmultiplications. Note that, for all a, b ∈ R and k, l ∈ Z , σ ′ (Φ( b ) − l Φ( v )) = Φ ∗ ( σ − ¯ v )(Φ( b ) − l Φ( v )) = Φ( σb + lσv − vb − lv ) , and, similarly, (Φ( a ) − k Φ( v )) σ ′ = Φ( aσ + kvσ − av − kv ) . Therefore, in view of the definition of s ′ in (3.19),Θ( a, k )Θ( b, l ) = ((Φ( a ) + k Φ( v ))(Φ( b ) + l Φ( v ))+ kσ ′ (Φ( b ) − l Φ( v )) + l (Φ( a ) − k Φ( v )) σ ′ + kls ′ , kl )= (Φ( ab + kvb + lav + klv + k ( σb + lσv − vb − lv )+ l ( aσ + kvσ − av − kv ) + kl ( s + v + v − vσ − σv )) , kl )= (Φ( ab + kσb + laσ + kls + klv ) , kl ) = Θ (( a, k ) ( b, l )) , as needed. Finally, setting Θ R = Φ we obtain the required weak equivalence of infinitehomothetic extensions (in the sense of Definition 3.1).For the statement (2), if ( ϕ R , R ( σ, s ) , ϕ Z ) and ( ϕ ′ R , R ( σ ′ , s ′ ) , ϕ ′ Z ) are equivalent byΘ, then we can follow arguments of the proof of (1) with Θ R = id R , to obtain thetranslational isomorphism of trussesΘ T ( a ) = a + e = [ a, , e ] = τ e ( a ) . Hence T ( σ, s ) tr ∼ = T ( σ ′ , s ′ ).If T ( σ, s ) tr ∼ = T ( σ ′ , s ′ ), then there exists e ∈ R such that τ e is an isomorphism of trussesand so, by the argument of the proof of Lemma 3.9 the corresponding isomorphism ofrings is Φ( a ) = τ e ( a ) − τ e (0) = a , and thusΘ : R ( σ, s ) −→ R ( σ ′ , s ′ ) , ( a, k ) ( a + kv, k ) , DEAL RING EXTENSIONS AND TRUSSES 15 is the corresponding equivalence of ring extensions.The cyclic homothetic extension case is treated in exactly the same way. (cid:3)
We note in passing that statement (2) of Theorem 3.11 can be viewed as transla-tional invariance of equivalence classes of ideal ring extensions by Z as modificationsof homothetic trusses by a translational isomorphism does not lead one out of theequivalence class of the corresponding extension.We end this section by studying T( σ, s ) in some special cases or with additionalproperties. Proposition 3.12.
Let ( σ, s ) be a homothetic datum on a ring R .(1) The truss T( σ, s ) is commutative if and only if R is commutative and σ is acentral double homothetism, that is, ← σ = → σ .(2) The truss T( σ, s ) has an absorber if and only if σ is an inner double homothetism.In this case, the ring retract of T( σ, s ) is isomorphic to R .(3) The truss T( σ, s ) has an identity if and only if T( σ, s ) ∼ = T(id , .(4) R has identity if and only if T(id , ∼ = T( R ) . In this case, T( σ, s ) ∼ = T( R ) .Proof. The truss T( σ, s ) is commutative if, and only if, for all a, b ∈ R , ab + aσ + σb = ba + bσ + σa. (3.24)Setting either a = 0 or b = 0 we obtain the centrality of σ and then the commutativity of R follows. In the converse direction (3.24) is obviously satisfied. This proves statement(1).An element e ∈ R is an absorber in T( σ, s ) if and only if, for all a ∈ R , ae + aσ + σe + s = e & ea + eσ + σa + s = e. (3.25)Setting a = 0 in (3.25) we obtain that σe + s = e and eσ + s = e , so plugging theseback into (3.25) we conclude that σ = − e . Conversely, if σ = ¯ b for some b ∈ R , thenthe equality (3.11b) implies that b = ¯ b + ¯ s and using this relation one easily checksthat equations (3.25) are satisfied with e = − b + s .If e is an absorber then, the map R −→ R , r r − e is the required isomorphismof rings R(T( σ, s ); e ) ∼ = R . This completes the proof of statement (2).An element u is the identity for T( σ, s ), if and only if, for all a ∈ R , au + aσ + σu + s = a = ua + uσ + σa + s. (3.26)Setting a = 0 we obtain s = − σu = − uσ , and thus the existence of the identity u implies that aσ = a − au & σa = a − ua. In other words, σ = id − ¯ u . Now setting a = u in (3.26), we obtain0 = s + u − u + uσ + σu, and hence Φ = id and v = − u induce the required isomorphism of trusses T( σ, s ) ∼ =T(id , v of R are suchthat σ = Φ ∗ (id − ¯ v ) = id − Φ( v ) and s = Φ( v + v − vσ − σv ) = Φ( v − v ) , then u = Φ( v ) is the identity in the truss T( σ, s ). Therefore, statement (3) holds.Finally, if R has the identity 1, then every homothetism is inner as σ = σ σ .Then setting v = σ σ, s ) ∼ = T(0 ,
0) = T( R ). In particular, T(id , ∼ = T( R ). Conversely, if thislast isomorphism holds, then id = − Φ( v ), for some v ∈ R and an automorphism Φ of R , which means precisely that − Φ( v ) is the identity for R . (cid:3) Proposition 3.13.
Let A be a ring with identity and B be any ring. Then any ho-mothetic truss on the product ring R = A × B is isomorphic to the product truss T( A ) × T( σ B , s B ) , for some homothetic datum on B .Proof. Let ( σ, s ) be a homothetic datum on R = A × B . Then, for all ( a, b ) ∈ R , σ ( a, b ) = ( → σ [1] ( a, b ) , → σ [2] ( a, b )) , ( a, b ) σ = ( ← σ [1] ( a, b ) , ← σ [2] ( a, b )) , for some additive functions → σ [1] , ← σ [1] : A × B −→ A , and → σ [2] , ← σ [2] : A × B −→ B .Since σ is a right operator, σ ( a,
0) = σ ( a, ,
0) = ( → σ [1] ( a, , → σ [2] ( a, ,
0) = ( → σ [1] ( a, , . Hence → σ [2] ( a,
0) = 0. Furthermore,(0 ,
0) = σ (0 ,
0) = σ (0 , b )(1 ,
0) = ( → σ [1] (0 , b ) , → σ [2] (0 , b ))(1 ,
0) = ( → σ [1] (0 , b ) , , which implies that → σ [1] (0 , b ) = 0. Therefore, σ ( a, b ) = σ ( a,
0) + σ (0 , b ) = ( → σ [1] ( a, , → σ [2] (0 , b )) . In a similar way, ( a, b ) σ = ( ← σ [1] ( a, , ← σ [2] (0 , b )) . We thus conclude that σ = ( σ A , σ B ) , where σ A is a double operator on A and σ B is a double operator on B given by σ A a = → σ [1] ( a, , aσ A = ← σ [1] ( a, , σ B b = → σ [2] (0 , b ) , bσ B = ← σ [2] (0 , b ) . Hence the problem of constructing homothetic trusses on R splits into the problemsof such constructions on A and B separately. Since A has the identity, by statement(4) in Proposition 3.12 all homothetic trusses A are isomorphic to T( A ), and thus theassertion follows. (cid:3) From trusses to ring extensions
In this section we start with a truss T and first assign a ring to it and then homotheticextension of this ring in which the truss is contained as in Theorem 3.6. This is achievedin two steps. First we associate a ring R( T ; e ) to any truss T (or, more generally aparagon) and any element e in this truss (not necessarily an absorber or a centralelement as in [Br19, Corollary 5.2] or [Br20, Lemma 3.14]). Next we show that thereis a homothetic datum ( σ, s ) on R( T ; e ) stemming from the internal structure of thetruss T such that the induced truss coincides with the original T , that is, T( σ, s ) = T . DEAL RING EXTENSIONS AND TRUSSES 17
Let ( T, [ − , − , − ] , · ) be a truss. Given an element e ∈ T , the induced actions of T onitself are defined as follows a e ⊲ b = λ e ( a, b ) := [ ab, ae, e ] , (4.1a) a e ⊳ b = ̺ e ( a, b ) := [ ab, eb, e ] , (4.1b)for all a, b ∈ T . As shown in [Br20] both λ e and ̺ e are heap homomorphisms inboth arguments, which means that both e ⊲ and e ⊳ distribute over the heap operation.Furthermore, λ e is a left action of the semigroup ( T, · ) while ̺ e is a right action. Inaddition the actions commute or satisfy the bimodule associative law, that is, for all a, b, c ∈ T , a e ⊲ ( b e ⊳ c ) = ( a e ⊲ b ) e ⊳ c. (4.2)Indeed, a e ⊲ ( b e ⊳ c ) = a e ⊲ [ bc, ec, e ] = [ a [ bc, ec, e ] , ae, e ] = [ abc, aec, e ] , by the distributive law and (2.1a)-(2.1b). On the other hand and by the same token,( a e ⊲ b ) e ⊳ c = [ ab, ae, e ] e ⊳ c = [[ ab, ae, e ] c, ec, e ] = [ abc, aec, e ] , as required. Therefore, we can write a e ⊲ b e ⊳ c for both ways of mixing the actions.Finally, the Mal’cev identities imply that e absorbs induced actions, that is, for all a ∈ T , a e ⊲ e = e e ⊳ a = e. (4.3) Definition 4.1.
Let ( T, [ − , − , − ] , · ) be a truss. A sub-heap S of ( T, [ − , − , − ]) is saidto be left-closed (respectively, right-closed ) if, there exists e ∈ S such that λ e ( S, S ) ⊆ S (respectively, ̺ e ( S, S ) ⊆ S ) . (4.4) Remark . Note that the existential quantifier in Definition 4.1 can be replaced bythe universal one. Indeed, if the condition (4.4) is satisfied for all e ∈ S = ∅ , then suchan e exists. Conversely, if (4.4) is satisfied for some e ∈ S , then for any e ′ , s, s ′ ∈ S , s e ′ ⊲ s ′ = [ ss ′ , se ′ , e ′ ] = [[ ss ′ , se, e ] , [ se ′ , se, e ] , e ′ ] = [ s e ⊲ s ′ , s e ⊲ e ′ , e ′ ] ∈ S, and similarly for the right action.Obviously T is both left and right closed. Similarly, if P is a left paragon in T ,then, by the definition, λ e ( T, P ) ⊆ P , for all e ∈ P , and hence it is left-closed. For thesymmetric reason, right paragon is right-closed. Theorem 4.3.
Let ( T, [ − , − , − ] , · ) be a truss. Let S be a left- or right-closed sub-heapof T . For any e ∈ S , the binary operation • e on S defined by, a • e b := [ a e ⊲ b, e e ⊲ b, e ] = [ a e ⊳ b, a e ⊳ e, e ] , (4.5) for all a, b ∈ S , makes the abelian group ( S, + e ) into an associative ring. We denotethis ring by R( S ; e ) .For all e, f ∈ T , consider the translation heap automorphism τ fe : T −→ T , a [ a, e, f ] ; see (3.23) . Let e ∈ S and let f ∈ T be such that λ e ( f, S ) ⊆ S (respectively, ̺ e ( S, f ) ⊆ S ) . (4.6) Then τ fe ( S ) is a left-closed (respectively, right-closed) sub-heap of T and τ fe restricts toan isomorphism of rings R( S ; e ) −→ R( τ fe ( S ); f ) .Proof. In order to avoid unwieldy expressions that are too hard to read with ease,in what follows we will suppress the indices e in expressions for products, sums andactions, and keep them only in places where an action induced by a different elementappears.First we check the equality of two expressions for • in equation (4.5). This followsby the application of the symmetry rule (2.2b),[ a ⊲ b, e ⊲ b, e ] = [ ab, ae, e, eb, e , e, e ] = [ ab, eb, e, ae, e , e, e ] = [ a ⊳ b, a ⊳ e, e ] . As a consequence of this equality the operation • is a binary operation on S in bothcases; if S is left-closed we use the left actions and when S is right-closed we use theright ones.The distributive law for • over + follows by the distributive laws of actions, by theabsorption rules (4.3) and the rearrangement rules (2.2). Explicitly, for all a, b, c ∈ S , a • ( b + c ) = a • [ b, e, c ] = [ a ⊲ [ b, e, c ] , e ⊲ [ b, e, c ] , e ]= [ a ⊲ b, e, a ⊲ c, e ⊲ b, e, e ⊲ c, e ]= [ a ⊲ b, e ⊲ b, a ⊲ c, e, e, e ⊲ c, e ]= [ a ⊲ b, e ⊲ b, e, e, a ⊲ c, e ⊲ c, e ] = a • b + a • c. The right distributive law follows by symmetry through expressing the multiplication • in terms of the right induced action. Finally, the associative law for • is a consequenceof the possibility of expressing of this operation in two different ways in (4.5) and thebimodule associative law (4.2). Explicitly, for all a, b, c ∈ S , a • ( b • c ) = [ a ⊲ ( b • c ) , e ⊲ ( b • c ) , e ]= [ a ⊲ b ⊳ c, a ⊲ b ⊳ e, a ⊲ e, e ⊲ b ⊳ c, e ⊲ b ⊳ e, e ⊲ e, e ]= [ a ⊲ b ⊳ c, a ⊲ b ⊳ e, e, e ⊲ b ⊳ c, e ⊲ b ⊳ e ]On the other hand,( a • b ) • c = [( a • b ) ⊳ c, ( a • b ) ⊳ e, e ]= [ a ⊲ b ⊳ c, e ⊲ b ⊳ c, e ⊳ c, a ⊲ b ⊳ e, e ⊲ b ⊳ e, e ⊳ e, e ]= [ a ⊲ b ⊳ c, e ⊲ b ⊳ c, e, a ⊲ b ⊳ e, e ⊲ b ⊳ e ]= [ a ⊲ b ⊳ c, a ⊲ b ⊳ e, e, e ⊲ b ⊳ c, e ⊲ b ⊳ e ] = a • ( b • c ) , as required.By [Br20, Proposition 4.28], τ fe is an isomorphism of T -modules ( T, ⊲ ) and ( T, f ⊲ ), aswell as T -modules ( T, ⊳ ) and ( T, f ⊳ ), that is, it is an isomorphism of heaps such that,for all a, b ∈ T , τ fe ( a ⊲ b ) = a f ⊲ τ fe ( b ) & τ fe ( b ⊳ a ) = τ fe ( b ) f ⊳ b. (4.7)In particular, it is an isomorphism of groups G( T ; e ) −→ G( T ; f ) and, hence, if e ∈ S ,it restricts to the isomorphism of groups G( S ; e ) −→ G( τ fe ( S ); f ). We need to show DEAL RING EXTENSIONS AND TRUSSES 19 that τ fe ( S ) is a closed sub-heap. Assume that f satisfies the first of conditions in (4.6).Then, for all a, b ∈ S , τ fe ( a ) ⊲ b = [ a ⊲ b, e ⊲ b, f ⊲ b ] = [ a • b, e, f ⊲ b ] ∈ S. Therefore, by the first of equations (4.7), τ fe ( a ) f ⊲ τ fe ( b ) = τ fe ( τ fe ( a ) ⊲ b ) ∈ τ fe ( S ) , and hence τ fe ( S ) is left-closed. If the other condition in (4.6) is satisfied, then we canuse the second of the module map properties (4.7) to draw the required conclusion.To complete the proof we only need to show that τ fe preserves the multiplications.To this end let us take any a, b ∈ S and compute, τ fe ( a ) • f τ fe ( b ) = [ τ fe ( a ) f ⊲ τ fe ( b ) , f f ⊲ τ fe ( b ) , f ]= [[ a, e, f ] f ⊲ τ fe ( b ) , f f ⊲ τ fe ( b ) , f ]= [ a f ⊲ τ fe ( b ) , e f ⊲ τ fe ( b ) , f f ⊲ τ fe ( b ) , f f ⊲ τ fe ( b ) , f ]= [ τ fe ( a ⊲ b ) , τ fe ( e ⊲ b ) , τ fe ( e )] = τ fe ([ a ⊲ b, e ⊲ b, e ]) = τ fe ( a • b ) , where we use the cancellation laws (2.2a) and the module map property (4.7) to derivethe fourth equality. Therefore, τ fe restricted to S is an isomorphism of rings as asserted. (cid:3) Since T is closed, there is a family of isomorphic rings R( T ; e ) labelled by elements e ∈ T . These rings are of the main interest in what follows. Corollary 4.4. (1) If e is an absorber in a truss T , then, for all a, b ∈ T , a • b = ab, in R( T ; e ) .(2) For any ring R , R(T( R ); 0) = R . Consequently, for all e ∈ R , R(T( R ); e ) ∼ = R .Proof. The first statement follows immediately from the definition of an absorber andthe multiplication • (and the Mal’cev identities), while the second one is a consequenceof the first one and the second statement of Theorem 4.3. (cid:3) Definition 4.5.
Let ( T, [ − , − , − ] , · ) be a truss and let e ∈ T . A subgroup I ≤ G( T ; e )is said to be left invariant (respectively, right invariant ) if λ e ( e, I ) ⊆ I (respectively, ̺ e ( I, e ) ⊆ I ). The set of all left invariant subgroups of G( T ; e ) is denoted by Linv( T ; e )(respectively, Rinv( T ; e ) for right invariant subgroups). Lemma 4.6.
Let ( T, [ − , − , − ] , · ) be a truss.(1) For all natural n , if I ∈ Linv( T ; e ) , then λ e ( e n , I ) ⊆ I (resp. if I ∈ Rinv( T ; e ) ,then ̺ e ( I, e n ) ⊆ I ).(2) Let I ≤ G( T ; e ) be a subgroup such that e ∈ I . Then I ∈ Linv( T ; e ) (respectively, I ∈ Rinv( T ; e ) ) if and only if eI ⊆ I (respectively, Ie ⊆ I ). Proof.
The first statement follows by the fact that λ e and ̺ e are actions of the semigroup( T, · ). For the second statement, I ∈ Linv( T ; e ), if and only if, for all x ∈ I there exist y ∈ I such that e ⊲ x = [ ex, e , e ] = y, that is ex = [ ex, e , e, e, e ] = [ y, e, e ] = y + e ∈ I, as required. (cid:3) Proposition 4.7.
Let ( T, [ − , − , − ] , · ) be a truss. Then P = ∅ is a left (respectively,right) paragon in T if and only if, for all q ∈ P , τ eq ( P ) is a left ideal in R( T ; e ) such that τ eq ( P ) ∈ Linv( T ; e ) (respectively, τ eq ( P ) is a right ideal in R( T ; e ) such that τ eq ( P ) ∈ Rinv( T ; e ) ).Proof. Assume first that P is a left paragon. Then, since a left paragon in T is thesame as an induced submodule of the left regular module T , τ eq ( P ) is a left paragonin T by [BR20b, Proposition 3.4]. Furthermore, since e ∈ τ eq ( P ) it is a subgroup ofG( T ; e ). The paragon property implies that τ eq ( P ) ∈ Linv( T ; e ), and, for the samereason, for all a ∈ T and x ∈ τ eq ( P ), a • x = [ a ⊲ x, e ⊲ x, e ] ∈ τ eq ( P ) . Hence, τ eq ( P ) is an invariant ideal in R( T ; e ).In the converse direction, assume that P ⊆ T is such that, for all q ∈ P , I := τ eq ( P )is an invariant left ideal in R( T ; e ). Then, for all a ∈ T and x ∈ I , a ⊲ x = [ a ⊲ x, e ⊲ x, e, e, e ⊲ x ] = [ a • x, e ⊲ x, e ] ∈ I, so I is a paragon in T . Since P = τ qe ( I ) = { [ x, e, q ] | x ∈ I } , it is a paragon as well by[BR20b, Proposition 3.4]. (cid:3) Presently, for any truss T we describe a homothetic extension of R( T ; e ) whichcontains T . Theorem 4.8.
Let T be a truss and e ∈ T . Define the double operator ε on the abeliangroup G( T ; e ) by εa = e ⊲ a, aε = a ⊳ e, (4.8) for all a ∈ T . Then(1) The pair ( ε, e ) is a homothetic datum on R( T ; e ) .(2) As trusses, T = T( ε, e ) .Proof. Since ε is given by truss actions that preserve e , both maps are group endomor-phisms of G( T ; e ). To prove that ε is a double homothetism the following propertiesneed to be checked, for all a, b ∈ T : e ⊲ ( a • b ) = ( e ⊲ a ) • b, (4.9a)( a • b ) ⊳ e = a • ( b ⊳ e ) , (4.9b) a • ( e ⊲ b ) = ( a ⊳ e ) • b, (4.9c) e ⊲ ( a ⊳ e ) = ( e ⊲ a ) ⊳ e. (4.9d) DEAL RING EXTENSIONS AND TRUSSES 21
We start by proving equation (4.9a). By the bimodule property (4.2), the definitionof multiplication • , and the distributivity of actions e ⊲ ( a • b ) = e ⊲ [ a ⊳ b, a ⊳ e, e ] = [ e ⊲ ( a ⊳ b ) , e ⊲ ( a ⊳ e ) , e ]= [( e ⊲ a ) ⊳ b, ( e ⊲ a ) ⊳ e, e ] = ( e ⊲ a ) • b, as required. The equality (4.9b) is proven by the same arguments (but using the otherequivalent definition of the product • ). The multiplier property (4.9c) is also provenby direct calculations. On one hand, a • ( e ⊲ b ) = [ a ⊲ ( e ⊲ b ) , e ⊲ ( e ⊲ b ) , e ]= [ ae ⊲ b, e ⊲ b, e ] = [ aeb, ae , e , e b, e ] , where we have used the fact that ⊲ is a left action and its definition as well as therearrangement and cancellation properties (2.2). On the other hand, using analogousproperties of the right action we find( a ⊳ e ) • b = [( a ⊳ e ) ⊳ b, ( a ⊳ e ) ⊳ b, e ]= [ e ⊳ eb, a ⊲ e , e ] = [ aeb, e b, e , ae , e ]= [ aeb, ae , e , e b, e ] = a • ( e ⊲ b ) , as required. The final double homothetism condition (4.9d) is a special case of thebimodule property (4.2).Directly by the definition of the actions, εe = e ⊲ e = [ e , e , e ] = e ⊳ e = e ε, hence the first of conditions (3.19) is satisfied. Furthermore, since ⊲ is the action, forall a ∈ T , ε a − εa = e ⊲ a − e ⊲ a = [ e ⊲ a, e ⊲ a, e ] = e • a. In a similar way aε − aε = [ a ⊳ e , a ⊳ e, e ] = a • e , hence ε = ε + e as required forthe second of conditions (3.19). This proves statement (1).Since the heap structure of a retract of a heap is equal to the original heap structure, T and T( ε, e ) are mutually equal as heaps. Let us denote by ◦ the product in T( ε, e ).Then, for all a, b ∈ T , a ◦ b = a • b + aε + εb + e = [ ab, ae, e, eb, e , e, ae, e , e, e, eb, e , e, e, e ]= [ ab, ae, e, eb, e , e, ae, e , eb ]= [ ab, ae, ae, eb, e, e, e , e , eb ] = ab, where the fourth and last equalities follow by a repetitive use of the cancellation rule(2.2a), while the fifth equality follows by the symmetry rule (2.2b) under the cyclicpermutation of odd indices (3 , , (cid:3) Definition 4.9.
Let T be a truss and e ∈ T , then the ring R( T ; e )( ε, e ) is called an infinite homothetic extension ring of T and is denoted by T ( e ). Similarly, the ringR( T ; e ) c ( ε, e ) (if it exists) is called a finite homothetic extension ring of T and isdenoted by T c ( e ). In view of the definition of the multiplication • in R( T ; e ) and since aε = ae − e and εb = eb − e in G( T ; e ), the product of ring T ( e ) built on the abelian group G( T ; e ) × Z has the explicit form( a, k )( b, l ) = ( ab + ( l − ae + ( k − eb + ( k − l − e , kl ) , (4.10)for all a, b ∈ T and k, l ∈ Z .The results of Part 2 can be summarised as the following two statements:(1) There is a one-to-one correspondence between isomorphism classes of trussesand weak equivalence classes of extensions of rings by Z . Furthermore, up totranslational isomorphism infinite homothetic trusses on a ring R are in one-to-one correspondence with equivalence classes of extensions of R by Z .(2) There is a two-way onto correspondence between trusses with a selected elementand rings with homothetic data given by( R, σ, s ) T( σ, s ) , ( T, e ) R( T, e ) . (4.11)In particular, every truss is a homothetic truss, that is, it is of the form T( σ, s )for some ring R and a homothetic datum on R . Conversely, every ring can beunderstood as a ring associated to a truss T with a chosen element e . Part Interpretation Universality of homothetic extensions of trusses
It has been explained in [BR20a] that there is a method of embedding of any truss T in a ring by appending T with an absorber (zero). This is based on extending thetruss multiplication T to the coproduct (direct sum) of T with the singleton truss { } , T ⊞ { } . Presently we review this procedure in brief.The heap T ⊞ { } consists of odd-length words in letters in T and 0 of the followingform: fix an element e ∈ T ,0 , a, a e , a e e . . . e, a e . . . e , a ∈ T ; (5.1)see [BR20a, Proposition 3.6]. Any word is identified with a word obtained by inde-pendent permutations of elements in odd positions or even positions (in concord with(2.2b)). The operation is by concatenation of words followed by the removal of anypairs of identical letters placed in consecutive positions and application of the heapoperation to any triples of consecutive elements of T . The multiplication in T ⊞ { } isdefined by the rules, for all a, b ∈ T , a · b = ab, a · · a = 0 , (5.2)and extended to the whole of T ⊞ { } by the truss distributivity. The rules (5.2) implythat 0 is the absorber in this truss, the multiplication (5.2) makes the abelian groupG( T ⊞ { } ; 0) into a ring. We denote this ring by T . Observe that any homomorphismof rings ϕ : R −→ R ′ as a function is the same as a homomorphism of correspondingtrusses T( ϕ ) : T( R ) −→ T( R ′ ), therefore whenever we write a composition of a trusshomomorphism ψ : T −→ T( R ) with ϕ : R −→ R ′ , ϕ ◦ ψ we think of T( ϕ ) ◦ ψ . DEAL RING EXTENSIONS AND TRUSSES 23
Lemma 5.1.
Let T be a truss. An extension T has the following universal property.For any ring R and a homomorphism of trusses ϕ : T −→ T( R ) there exists a uniquering homomorphism ϕ b : T −→ R rendering commutative the following diagram T ι T / / ϕ ! ! ❉❉❉❉❉❉❉❉❉ T( T ) ∃ ! T( b ϕ ) z z ✈ ✈ ✈ ✈ ✈ T( R ) , where ι T : T −→ T( T ) is given by t t . A pair ( T , ι T ) is a universal arrow (see [Ma98, Section III.1, Definition] ).Proof. Let us consider the following commutative diagram of morphisms of trusses:T( R ) T ι T / / ϕ ♣♣♣♣♣♣♣♣♣♣♣♣♣ T( T ) e ϕ O O ✤✤✤ { } , T( ι ) o o T( j ) g g ❖❖❖❖❖❖❖❖❖❖❖❖❖ (5.3)where j and ι are unique ring homomorphisms from the zero object { } in the categoryof rings. The existence of the unique truss morphism e ϕ : T( T ) = T ⊞ { } −→ T( R )follows by the universal property of the coproduct. Since e ϕ ◦ T( ι )(0) = e ϕ (0) = T( j )(0) = 0 R , e ϕ = T( ϕ b ) for some (unique) ring homomorphism ϕ b : T −→ R . (cid:3) The following theorem allows one to identify the universal ring extension T with aninfinite homothetic ring extension T ( e ) of Definition 4.9. Theorem 5.2.
Let T be a truss and e ∈ T . For any a ∈ T and n ∈ Z define thefollowing elements of T ⊞ { } : a [ n ] = a e e . . . e | {z } n − , n > a e e . . . e | {z } − n +3 , n ≤ . (5.4) Then:(1) T ⊞ { } = { a [ n ] | a ∈ T, n ∈ Z } .(2) The map χ e : T −→ T ( e ) , a [ n ] ( a, n ) , is a unique isomorphism of rings such that χ e ( a ) = ( a, .Proof. (1) Note that e [0] = e e e [0] = 0, a [ n ], n ≥
0, describe all theelements of the first four types listed in (5.1) (in particular a [1] = a ). Finally,0 a e . . . e e e a e . . . e e a e e . . . e
0= [ e, a, e ] e e e . . . e e, a, e ][ n ] , for the negative n such that − n + 1 is equal to the length of the original word. Thiscompletes the proof of statement (1).(2) In view of the assertion of statement (1) and since there are no repetitions ofelements listed as a [ n ], it is clear that the map χ e is a bijection. By the universalproperty of coproduct it is a unique homomorphism of heaps that fits the diagramH( T ( e )) T ι T / / ι T ( e ) ♦♦♦♦♦♦♦♦♦♦♦♦♦♦ T ⊞ { } χ e O O ✤✤✤ { } , ι o o j h h PPPPPPPPPPPPP (5.5)where ι T : a a = a [1], ι : 0 e [0], j : 0 ( e,
0) and ι T ( e ) : a ( a, χ e (0) = ( e,
0) so the zero of the ring T is transformed to the zero of T ( e ).Hence by the Lemma 5.1 the map χ e is a (unique) homomorphism of rings.The additivity of χ e can also be checked directly by considering the following fourcases. For positive k, l ∈ Z , a [ k ] + b [ l ] = a e e . . . e | {z } k − b e e . . . e | {z } l − = a b e . . . e | {z } k + l ) − = a e e b e . . . e | {z } k + l )+1 = a e b e e . . . e | {z } k + l )+1 = [ a, e, b ] 0 e e . . . e | {z } k + l ) − = [ a, e, b ][ k + l ] , where we have used the symmetry to swap e with b and other rules of operations in T ⊞ { } . Therefore, χ e ( a [ k ] + b [ l ]) = ([ a, e, b ] , k + l ) = ( a + e b, k + l ) = ( a, k ) + ( b, l ) = χ e ( a [ k ]) + χ e ( b [ l ]) . The case of two non-positive indices is treated in a similar way. Now assume that k > l ≤ k + l >
1. Then, for all a, b ∈ T , a [ k ] + b [ l ] = a e e . . . e | {z } k − b e e . . . e | {z } − l +3 = a e e . . . e | {z } k − e e . . . e b | {z } − l +3 = a e e . . . b | {z } k + l ) − = a b e e . . . | {z } k + l ) − = a e e b e . . . e | {z } k + l )+1 = [ a, e, b ] 0 e . . . e | {z } k + l ) − = [ a, e, b ][ k + l ] , where the second, fourth and sixth equalities use the freedom of swapping elements inpositions with matching parities and the remaining equalities use the cancellation ofrepeated letters and the application of the heap operation in T . Therefore, χ e ( a [ k ] + b [ l ]) = χ e ( a [ k ]) + χ e ( b [ l ]) also in this case. The case of k + l ≤ χ e is an isomorphism of abelian groups. DEAL RING EXTENSIONS AND TRUSSES 25
To check directly that χ e preserve multiplication observe that in the ring T , for all a, b ∈ T and k, l ∈ Z , a [1] · b [1] = ab [1] , a [ k ] + b [ l ] = ( a + e b )[ k + l ] , a [ k ] = a [1] + ( k − e [1] , (5.6)since a [1] = a , + = + in T and by the additivity of χ e . Here the concatenation ab means the product in T . Using (5.6) and with the understanding that + in-betweenelements of T means + e (i.e. the operation in the retract G( T ; e )), while + in-betweenelements of T means + , we can thus compute, a [ k ] · b [ l ] = ( a [1] + ( k − e [1]) · ( b [1] + ( l − e [1])= ab [1] + ( l − ae [1] + ( k − eb [1] + ( k − l − e [1]= ( ab + ( l − ae + ( k − eb + ( k − l − e )[ kl ]= ( ab + ( l − ae + ( k − eb + ( k − l − e )[ kl ] . Hence, χ e ( a [ k ] · b [ l ]) = ( ab + ( l − ae + ( k − eb + ( k − l − e , kl )= ( a, k )( b, l ) = χ e ( a [ k ]) χ e ( b [ l ]) , by (4.10). Therefore, χ e is an isomorphism of rings as stated. (cid:3) The identification of the homothetic extension ring T ( e ) with the ring T allows oneto reveal the universality of the former. Corollary 5.3.
For any truss T and e ∈ T , the infinite homothetic extension ring T ( e ) has the same universal property as T . That is, for any ring R and a homomorphismof trusses ϕ : T −→ T( R ) there exists a unique ring homomorphism b ϕ : T ( e ) −→ R such that T( b ϕ ) ◦ ι T ( e ) = ϕ. Lemma 5.4.
Let T be a truss and e ∈ T . The truss homomorphism ι T ( e ) : T −→ T( T ( e )) has the following cancellation property. For all truss homomorphisms ϕ, ψ :T( T ( e )) −→ U such that ϕ ( e,
0) = ψ ( e, , ϕ ◦ ι T ( e ) = ψ ◦ ι T ( e ) implies ϕ = ψ. In particular, if U = T( R ) for a ring R , then for all ring homomorphisms f, g : T ( e ) −→ R , T( f ) ◦ ι T ( e ) = T( g ) ◦ ι T ( e ) implies f = g. Proof.
Recall that elements of T ( e ) are of the form ( a, k ) ∈ T × Z . We will provethe lemma by induction on k (separately for positive and negative integers). Let ϕ, ψ : T( T ( e )) −→ U be truss morphisms such that ϕ ( u,
0) = ψ ( u, u ∈ T ,and ϕ ◦ ι T ( e ) = ψ ◦ ι T ( e ) . The second condition means that ϕ ( a,
1) = ψ ( a, a ∈ T . Assume that ϕ ( a, k ) = ψ ( a, k ) for some positive k ∈ Z and all a ∈ T . Then, ϕ ( a, k + 1) = ϕ ([ a, u, u ] , k − ϕ ([( a, k ) , ( u, , ( u, ϕ ( a, k ) , ϕ ( u, , ϕ ( u, ψ ( a, k ) , ψ ( u, , ψ ( u, ψ ( a, k + 1) , since both ϕ and ψ are heap homomorphisms. Similarly, for all k ≤
1, if ϕ ( a, k ) = ψ ( a, k ), then ϕ ( a, k −
1) = ϕ ([ a, u, u ] , k − ϕ ([( a, k ) , ( u, , ( u, ϕ ( a, k ) , ϕ ( u, , ϕ ( u, ψ ( a, k ) , ψ ( u, , ψ ( u, ψ ( a, k − . Therefore, ϕ = ψ . Finally, since ( e,
0) is the zero of the ring T ( e ) the first condition issatisfied with u = e and hence the second assertion follows. (cid:3) Since, for every e ∈ T , T ( e ) is isomorphic to T , the truss homomorphism ι T : T −→ T( T ) defined in Lemma 5.1 has a similar cancellation property: Corollary 5.5.
For all trusses T and for all ring homomorphisms f, g : T −→ R , if T( f ) ◦ ι T = T( g ) ◦ ι T , then f = g .Proof. If T( f ) ◦ ι T = T( g ) ◦ ι T then T( f ) ◦ ι T ◦ T( χ − e ) = T( g ) ◦ ι T ◦ T( χ − e ), where χ e : T −→ T ( e ) is the ring isomorphism constructed in Theorem 5.2. Since ι T ◦ T( χ − e ) = ι T ( e ) by the diagram (5.5), the assertion follows from Lemma 5.4. (cid:3) The universal property of the ring T described in Lemma 5.1 gives rise to a functor( − ) : Trs −→ Ring between categories of trusses and rings, see [Ma98, Section IV,Theorem 2(ii)]. For the sake of keeping the presentation self-contained, we add a shortproof. The functor is given for all trusses T by T T , and for all morphisms ϕ ∈ Hom
Trs ( T, U ) by ϕ ϕ := \ ι U ◦ ϕ , where b denotes the ring homomorphisminduced from a truss homomorphism via the diagram in Lemma 5.1. Observe that, byLemma 5.1, for all ϕ ∈ Hom
Trs ( T, U ) and ψ ∈ Hom
Trs ( U, V ),T( ψ ◦ ϕ ) ◦ ι T = T( \ ι V ◦ ψ ) ◦ T( \ ι U ◦ ϕ ) ◦ ι T = T( \ ι V ◦ ψ ) ◦ ι U ◦ ϕ = ι V ◦ ψ ◦ ϕ = T( \ ι V ◦ ψ ◦ ϕ ) ◦ ι T = T(( ψ ◦ ϕ ) ) ◦ ι T . Lemma 5.4 implies that ψ ◦ ϕ = ( ψ ◦ ϕ ) . Thus the composition is preserved by the assignment. One can easily check that identitymorphisms are preserved. Hence, ( − ) : Trs −→ Ring is a functor.The following proposition follows by [Ma98, Section IV, Theorem 2(ii)], but for thesake of the unaccustomed reader, we sketch a proof.
Proposition 5.6.
The functor ( − ) is left adjoint to the functor T :
Ring −→ Trs .Proof.
For all trusses T and rings R let us consider the functions α T,R : Hom
Ring ( T , R ) −→ Hom
Trs ( T, T( R )) , f T( f ) ◦ ι T . We will show that these functions define a natural isomorphism of bifunctors α :Hom Ring (( − ) , − ) −→ Hom
Trs ( − , T( − )).The functions α T,R are injective by Corollary 5.5. The universal property in Lemma 5.1immediately implies that the α T,R are also onto. For naturality, take any rings R , S and trusses T , U , and consider homomorphisms f : T −→ R , ϕ : U −→ T and g : R −→ S . Then α U,R ( f ◦ ι T ◦ ϕ b ) = T( f ◦ ι T ◦ ϕ b ) ◦ ι U = T( f ) ◦ ι T ◦ ϕ = α T,R ( f ) ◦ ϕ, by Lemma 5.1. Similarly, α T,S ( g ◦ f ) = T( g ◦ f ) ◦ ι T = T( g ) ◦ α T,R ( f ) , as T( g ) = g as functions. Therefore α is a natural isomorphism and the extension torings functor ( − ) is the left adjoint to T. (cid:3) DEAL RING EXTENSIONS AND TRUSSES 27
Combining Proposition 5.6 with Theorem 5.2 we thus obtain
Corollary 5.7.
For all rings R and trusses T , and for all e ∈ T , Hom
Ring ( T ( e ) , R ) ∼ = Hom Trs ( T, T( R )) . Minimality of homothetic extensions of trusses
In Section 5 we have described the universal property of the ring T ( e ) and shownthat the assignment of the ring T to a truss T is functorial. It seems quite naturalthat when one extends a truss T to a ring, one would like to obtain as “small” a ringas possible. Therefore, in this section we will use the universal property to describedifferent kinds of “smallness” of truss extensions into rings. To fix notation, wheneverwe write ϕ b we think of a unique filler of the diagram from Corollary 5.3. Definition 6.1.
Let T be a truss, R a ring and let η R : T −→ T( R ) be an injectivehomomorphism of trusses. We say that R is a locally small extension of T if there isno subring S ( R such that η R ( T ) ⊆ S . Proposition 6.2.
Let T be a truss and R be an extension of T into a ring with injection η R : T −→ T( R ) . Then R is a locally small extension if and only if R = Im( η R b ) ∼ = T ( e ) / ker( η R b ) , for all e ∈ T .Proof. Let us assume that R is a locally small extension of T with η R : T −→ T( R )and take any e ∈ T . By Corollary 5.3, there exists a unique ring homomorphism η R b : T ( e ) −→ R such that T( η R b ) ◦ ι T ( e ) = η R . Consequently, S = Im( η R b ) is a subringof R such that η R ( T ) ⊆ S , and hence S = R , by the local smallness of the extension R . The first isomorphism theorem for rings yields the required isomorphism.In the converse direction, let R = Im( η R b ) (or, equivalently, R ∼ = T ( e ) / ker( η R b )) andsuppose that there is a subring S of R such that η R ( T ) ⊆ S . Let j : S −→ R be theinclusion ring homomorphism and let η S : T −→ S be given by T ( j ) ◦ η S = η R . Allthese maps together with the corresponding ring homomorphisms c η R and b η S can befitted in the commutative diagram: T ι T ( e ) / / η S ! ! ❉❉❉❉❉❉❉❉❉ η R ' ' T ( e ) T( c η S ) { { ✇✇✇✇✇✇✇✇✇ T( c η R ) v v T( S ) T( j ) (cid:15) (cid:15) T( R ) . Hence j ◦ η S b = η R b , which implies that R = Im( η R b ) ⊆ S , that is, S = R . Therefore, R is a locally small ring extension of T . (cid:3) Remark . Proposition 6.2 indicates that a locally small extension of a truss T intoa ring is not necessarily unique (not even up to isomorphism) and also provides onewith a method of constructing such extensions. One needs simply to take any ring R which embeds T as a sub-truss of T( R ) via an inclusion map, say, η R , construct thecorresponding unique ring homomorphism c η R : T ( e ) −→ R . The ring S = Im( c η R ) ⊆ R together with the corestriction of η R to S is the required locally small extension (notethat η R ( T ) ⊆ S , since c η R ◦ ι T ( e ) = η R ). The form of the ring map c η R : T ( e ) −→ R canbe easily worked out by inductive arguments. Explicitly, c η R : ( a, k ) η R ( a ) + ( k − η R ( e ) . In particular, the map ι T ( e ) b corresponding to the canonical truss inclusion ι T ( e ) : T −→ T ( e ) is equal to the identity map, and hence T ( e ) is a locally small extension of T . Inview of Proposition 6.2 all other locally small extensions in T correspond to suitableideals in T ( e ). Example 6.4.
For any integer r ≥ r = − Z ), T r = r ( r − Z + r = { r (( r − k + 1) | k ∈ Z } . Note that the multiplication in T r is well-defined since r is congruent to r modulo r ( r − Z ), with the embedding η : n n .The map η induces a homomorphism of rings b η : T r ( r ) Z , which in view ofRemark 6.3 reads b η ( r ( r − k + r, l ) = r (( r − k + l ) , for all k, l ∈ Z . Hence the ring r Z = Im( b η ) is a locally small ring extension of T r . SinceIm( b η ) ( Z , Z is not a locally small extension of T r for all r = − T correspond to certainideals I in T ( e ). By Theorem 3.6(3), ι T ( e ) ( T ) is a paragon in T ( e ) and thusI( T ) := τ ( e, e, ( ι T ( e ) ( T )) = { ( a, | a ∈ T } is an ideal in the homothetic extension T ( e ). To ensure that the composite map T (cid:31) (cid:127) / / I( T ) (cid:31) (cid:127) / / T ( e ) / / / / T ( e ) /I , is an injective map, we need to require that I intersects trivially with I( T ). In summary,we can state Lemma 6.5.
Let T be a truss, I be an ideal in T ( e ) and π : T ( e ) −→ T ( e ) /I be acanonical epimorphism. Then T ( e ) /I is a locally small extension of T into a ring withan injection π ◦ ι T ( e ) : T −→ T ( e ) /I if and only if I ∩ I( T ) = { ( e, } . Corollary 6.6.
Let T be a truss such that the (any) retract G( T ; e ) has a finite ex-ponent. Then the cyclic homothetic extension T c ( e ) is a locally small extension of T .Proof. Let N be the exponent of G( T ; e ), and let I N = { ( e, N k ) | k ∈ Z } be theideal of T ( e ) that defines the cyclic homothetic extension T c ( e ) = T ( e ) /I N . Then( e, N k ) ∈ I( T ) if and only if k = 0, and hence T c ( e ) is a locally small extension asstated. (cid:3) In the hierarchy of locally small extensions of a truss T one can distinguish thosethat are particularly close to T . DEAL RING EXTENSIONS AND TRUSSES 29
Definition 6.7.
A locally small extension (
S, η S ) of a truss T is called a small extension provided η S b (I( T )) is an essential ideal in S .Taking into account the explicit form of the induced ring map η S b described in Re-mark 6.3 one immediately obtains the following characterisation of small extensions. Lemma 6.8.
Let T be a truss, e ∈ T and let ( S, η S ) be a locally small extension of T .Then ( S, η S ) is a small extension if and only if, for all ideals J ✁ S , there exists a ∈ T such that a = e and η S ( a ) − η S ( e ) ∈ J . Example 6.9.
Let T r be the truss defined in Example 6.4. Consider the locally smallextension η : T r −→ T( r Z ), n n (with e = r ). Since r Z is a principal ideal domain,all ideals in r Z are of the form I q = qr Z , for a non-negative integer q . Then I q ∋ qr ( r −
1) = η ( r ( r − q + r ) − η ( r ) , and hence r Z is a small extension of T r .There exist locally small extensions which are not small extensions. As we observedin Remark 6.3, T ( e ) is a locally small extension of a truss T but usually it is not small.For example if T = T( R ), for some ring R , then T( R )(0) = R × Z and clearly ( R, R × Z . Similarly, if ( R, +) has a finite exponent N , thenT( R ) c (0) = R × Z N and e.g. (0 , Z N ) intersects trivially with ( R, Definition 6.10.
Let T be a truss and ( S, η S ) be a locally small extension of T into aring S . Then we say that ( S, η S ) is a minimal extension if, for all ideals I ⊆ T ( e ) suchthat I ∩ I( T ) = { ( e, } , I ⊆ ker( η S b ). Lemma 6.11.
A minimal extension of a truss is unique up to isomorphism.Proof.
Let (
S, η S ) and ( S ′ , η S ′ ) be two minimal extensions of a truss T . Then ker( η S b ) ⊆ ker( η S ′ b ) and ker( η S ′ b ) ⊆ ker( η S b ), so ker( η S b ) = ker( η S ′ b ), and isomorphism is given by thefirst isomorphism theorem for rings. (cid:3) Example 6.12 (Minimal extensions exist) . Let T = T( R ) for a ring R , then R is aminimal extension of T( R ) to a ring. Since T( R )(0) = R × Z , and the ring homomor-phism b η : R × Z −→ R induced from the identity map η : R −→ R is the projectionon the first factor. Furthermore, I( T ) = ( R, I ✁ R × Z intersectstrivially with ( R,
0) then I ⊆ (0 , Z ) = ker b η .Observe that if ( S, η S ) is a small extension, then it is a locally small extension. Ina similar way every minimal extension is a small extension. This easily follows by thedefinitions of locally small, small and minimal extensions. Lemma 6.13.
Let ( S, η S ) be an extension of a truss T with at least two elements suchthat S is a domain. If ( S, η S ) is a small extension, then ( S, η S ) is a minimal extension.Proof. Let I be an ideal in T ( e ) such that I ∩ I( T ) = { ( e, } . If ( a, k ) ∈ I , then forall b ∈ T , ( b, a, k ) ∈ I ∩ I( T ) , and hence ( b, a, k ) = ( e, η S b is a ring homomorphism, η S b ( b, η S b ( a, k ) = η S b ( e,
0) = 0 , and the fact that S is a domain implies that η S b ( b,
0) = 0 or η S b ( a, k ) = 0 , for all( a, k ) ∈ I and b ∈ T . In particular, for b = e , η S b ( b,
0) = η S ( b ) − η S ( e ) = 0 by theformula in Remark 6.3 and since η S is injective. Therefore η S b ( a, k ) = 0, for all ( a, k ) ∈ I and thus I ⊂ ker( η S b ) and ( S, η S ) is a minimal extension. (cid:3) Example 6.14.
Let us consider trusses T r = r ( r − Z + r , for r = − r ≥ T r ֒ → T( r Z ) is a small extension and since r Z is adomain, r Z is a minimal extension of T r by Lemma 6.13. Example 6.15 (Small but not a minimal extension) . Let p be a prime number andconsider the truss T = (cid:18) Z p (cid:19) with the usual matrix multiplication and the heap structure arising from the matrixaddition. For e = (cid:18) (cid:19) , the integral extension T ( e ) can be identified with T ( e ) = (cid:26)(cid:18) m a m (cid:19) | m ∈ Z , a ∈ Z p (cid:27) , a Dorroh extension of the ring Z p with zero multiplication. With this identification, ι T : T −→ T ( e ) is the obvious (set-theoretic) inclusion map, and the correspondingideal I( T ) of T ( e ) comes out as I( T ) = (cid:18) Z p (cid:19) . For all n ∈ N , let us define injective truss homomorphisms η n : T −→ T( Z p n +1 ) , (cid:18) a (cid:19) (1 − ap n ) (mod p n +1 ) . The universally constructed ring homomorphisms are b η n : T ( e ) −→ Z p n +1 , (cid:18) m a m (cid:19) ( m − ap n ) (mod p n +1 ) . Each of the maps b η n is onto so the extensions η n : T −→ Z p n +1 are locally small.Furthermore, since for all n , the ideals b η n (I( T )) = { ap n (mod p n +1 ) | a ∈ Z p } , are essential in Z p n +1 , all these extensions are small. By the uniqueness of the minimalextensions at most one of them could be minimal. Thus we obtain an infinite familyof small extensions that are not minimal. DEAL RING EXTENSIONS AND TRUSSES 31
Part Classifications Trusses from rings with zero multiplication
The results of Section 3 allow one to associate a truss to a homothetic datum ona ring, and thus provide one with a way of constructing trusses. In this part we willclassify or describe all trusses induced by homothetic data on rings with particularproperties. We start with the simplest possible rings, those with zero multiplication.
Proposition 7.1.
Let R be a ring with zero multiplication. Then:(1) Any homothetic datum on R consists of an element s ∈ R and a double operator σ such that(a) for all a ∈ R , σ ( aσ ) = ( σa ) σ ,(b) σ is an idempotent, that is σ = σ ,(c) sσ = σs .(2) Any truss induced by a homothetic datum is isomorphic to T( σ, , where σ sat-isfies conditions (a) and (b) above.(3) Two trusses T( σ, and T( σ ′ , are isomorphic if an only if there exists anabelian group automorphism Φ : R −→ R such that σ ′ = Φ ∗ ( σ ) .Proof. Since R has zero multiplication, all the bimultiplication conditions (3.3) areautomatically satisfied, so only (3.4) remains, and this is precisely condition (1)(a) inthe statement of the proposition. For the same reason, ¯ s is the zero operation, so thehomothetic datum conditions reduce to (1)(b) and (1)(c) above. Any s ∈ R can bereduced to zero by choosing Φ = id and v = 2 σs − s = 2 sσ − s in (3.19). Indeed, inthis case, s ′ = s + v + v − vσ − σv = s + 2 σs − s − σsσ + sσ − σ s + σs = 0 , by properties (1b) and (1c) and since R is a ring with zero multiplication. The assertion(3) follows immediately from Lemma 3.8 and Lemma 3.9. (cid:3) Our aim in this section is to reveal the contents of Proposition 7.1 in a way thatcould lead to the full classification of trusses built on rings with zero multiplication.Recall first that a double operator σ : R −→ R can be identified with an orderedpair of additive endomaps → σ and ← σ on R . Put together, conditions (1a) and (1b) inProposition 7.1 mean that → σ and ← σ are commuting idempotents in the endomorphismring End( R, +). The following lemmas are probably well known. We include them forcompleteness. Lemma 7.2.
Let S be a ring and e, f ∈ S be idempotent elements. The followingstatements are equivalent:(i) ef = f e ,(ii) there exists exactly one triple of orthogonal idempotents ( e , e , e ) in S such that e = e + e & f = e + e . (7.1) Proof.
If the idempotents e and f commute, then setting e = e − ef, e = f − f e, e = ef, we obtain a triple of orthogonal indempotent that satisfies (7.1). Suppose ( f , f , f )is another such triple. Then, since f = ef = f e , f = e − ef = e & f = f − f e = e , which proves the uniqueness.In the converse direction the orthogonality and idempotent property of the e i implythat e and f are idempotents and that ef = f e , as required. (cid:3) Lemma 7.3.
For any abelian group A , there is a bijective correspondence between thefollowing sets of data:(i) ordered pairs ( → σ , ← σ ) of commuting idempotents in the ring End( A, +) ;(ii) ordered triples ( ε , ε , ε ) of orthogonal idempotents in End( A, +) ;(iii) ordered quadruples ( A , A , A , A ) of subgroups of A such that A = A ⊕ A ⊕ A ⊕ A . Proof.
The equivalence of statements (i) and (ii) is proven in Lemma 7.2. Given systemof orthogonal idempotents in (ii) set ε = id − ε − ε − ε , and then define A i = Im ε i , i = 1 , . . . ,
4. Conversely, given ordered direct sum decomposition of A as in (iii), setthe ε i , i = 1 , , A i . (cid:3) Remark . There is a freedom in setting up ordering of tuples in Lemma 7.3. In thefollowing examples we choose the convention in which(a) → σ is a projection on A ⊕ A ,(b) ← σ is a projection on A ⊕ A .As a consequence of this choice:(c) → σ ◦ ← σ is a projection on A ,(d) → σ − → σ ◦ ← σ is a projection on A ,(e) ← σ − ← σ ◦ → σ is a projection on A ,(f) A = ker → σ ∩ ker ← σ .The identification in (f) follows from the fact that both → σ and ← σ are sums of orthogonalprojections.With all these results at hand we can now describe all trusses corresponding to ringswith zero multplication. Theorem 7.5.
Let A be an abelian group.(1) For any ordered quadruples A := ( A , A , A , A ) of abelian subgroups of A suchthat A = A ⊕ A ⊕ A ⊕ A , and any elements s ∈ A ⊕ A , the multiplication ( a + a + a + a )( b + b + b + b ) = b + a + a + b + s, (7.2) for all a i , b i ∈ A i , defines a truss on H( A ) . We denote this truss by T( A , s ) . DEAL RING EXTENSIONS AND TRUSSES 33 (2) Any truss T( A , s ) is isomorphic to T( A ) := T( A , , and T( A ) ∼ = T( B ) if andonly if A i ∼ = B i , i = 1 , . . . , .(3) The product in the homothetic extension T( A )(0) of the truss T( A ) is given bythe following formula, ( a + a + a + a , k ) ( b + b + b + b , l ) = ( k ( b + b ) + l ( a + a ) , kl ) , (7.3) for all a i , b i ∈ A i and k, l ∈ Z .(4) Any truss arising from or leading to a ring with zero multiplication (in the senseof the correspondence (4.11) ) is of the form T( A , s ) .Proof. If A is equipped with zero multiplication, by Proposition 7.1 and Lemma 7.3 inconjunction with Remark 7.4, the pair ( A , s ) gives a homothetic datum on this ring.The truss induced by this datum has multiplication (7.2) since σ ( b + b + b + b ) = → σ ( b + b + b + b ) = b + b , ( a + a + a + a ) σ = ← σ ( a + a + a + a ) = a + a . Hence T( A , s ) is a truss, as claimed. The remaining statements follow immediatelyfrom Proposition 7.1, the product formula (4.10), and the correspondence (4.11). (cid:3) Example 7.6.
Let A be a nontrivial indecomposable abelian group. Since it cannotbe written as a direct sum of two non-trivial subgroups, there are four possible orderedquadruples A that necessarily contain one copy of A and three copies of the trivialgroup 0. Multiplications in the corresponding trusses and their homothetic extensionsare collected in the following table: A Truss T( A ) Extension T( A )(0) ∀ a, b ∈ T ∀ a, b ∈ T, k, l ∈ Z ( A, , , ab = b ( a, k )( b, l ) = ( kb, kl )(0 , A, , ab = a ( a, k )( b, l ) = ( la, kl );(0 , , A, ab = a + b ( a, k )( b, l ) = ( la + kb, kl )(0 , , , A ) ab = 0 ( a, k )( b, l ) = (0 , kl ) Example 7.7.
Let p be a prime number and n be any natural number and set A tobe the abelian group A = Z np = Z p ⊕ Z p ⊕ . . . ⊕ Z p | {z } n -times . Since Z p is a simple cyclic group all subgroups of A are isomorphic to Z kp , 0 ≤ k ≤ n .Therefore, up to isomorphism, there are as many ordered partitions of A into thedirect sum of four subgroups as there are elements n = ( n , n , n , n ) ∈ N such that n + n + n + n = n . The corresponding groups are A i ∼ = Z n i p and they are uniquelydetermined by n . Hence, by Theorem 7.5 there are exactly (cid:0) n +33 (cid:1) non-isomorphictrusses T( A ). In view of the formula (7.2) the product in T( A ) comes out as, for all a = ( a i ) ni =1 , b = ( b i ) ni =1 ∈ Z np , ab = ( a + b , . . . , a n + b n , b n +1 , . . . b n + n , a n + n +1 , . . . , a n + n + n , , . . . , n X i =1 a i e i + n + n X i =1 b i e i + n + n + n X i = n + n +1 a i e i , where the e i are members of the standard basis for the Z p -vector space Z np . Therefore,the product in the infinite homothetic extension T( A )(0) is( a , k )( b , l ) = n X i =1 la i e i + n + n X i =1 kb i e i + n + n + n X i = n + n +1 la i e i , kl ! . The ring T( A )(0) can be identified with a particular subring of the ring M ( Z np ) of2 × Z np as follows. Set u = (1 , . . . , | {z } n + n , , . . . , | {z } n + n ) , v = (1 , . . . , | {z } n , , . . . , | {z } n , , . . . , | {z } n , , . . . , | {z } n ) . The collection of all upper-triangular matrices U := (cid:26)(cid:18) k u a k v (cid:19) | a ∈ Z np , k ∈ Z (cid:27) is a subring in M ( Z np ), since both u and v are idempotents. The functionT( A )(0) −→ U, ( a , k ) (cid:18) k u a k v (cid:19) , is the required isomorphism of rings.Since the abelian group Z np has the exponent p , the cyclic version of the homotheticextension exists. Its description is obtained by replacing Z by Z p in the above formulae.8. Trusses from rings with trivial annihilators
For a ring R we denote by a ( R ) = { a ∈ R | ∀ r ∈ R, ra = ar = 0 } , the annihilator ideal of R . In this section we describe all trusses that can be associ-ated to a ring with the trivial annihilator. We start by gathering some properties ofhomothetisms in this case (the first two are well known, see e.g. [An11, Section 2]). Lemma 8.1.
Let R be a ring such that a ( R ) = 0 . Then(1) The map β : R −→ Ω( R ) , a ¯ a, is injective (a monomorphism of rings).(2) Π( R ) = Ω( R ) .(3) For all σ ∈ Ω( R ) and s ∈ R such that σ = σ + ¯ s , sσ = σs. DEAL RING EXTENSIONS AND TRUSSES 35
Proof. If a, c ∈ R are such that ¯ a = ¯ c , then, for all b ∈ R ,( a − c ) b = 0 & b ( a − c ) = 0 , and thus a = c since a ( R ) = 0. This proves statement (1). For any σ ∈ Ω( R ) and a ∈ R , conditions (3.3) imply that ( σa ) σ = σ ( aσ ) , and hence (3.4) follows by assertion (1), and thus every bimultiplication is self-permutable,i.e. a homothetism, as required.Take σ and s such that σ − σ = ¯ s . Then starting with ¯ s and using this relationsufficiently many times one finds that ¯ sσ = σ ¯ s . The assertion (3) then follows by (3.7)and assertion (1). (cid:3) Theorem 8.2.
Let R be a ring such that a ( R ) = 0 . There is a bijective correspondencebetween isomorphism classes of homothetic trusses on R and equivalence classes ofidempotents in the ring Ξ( R ) of outer bimultiplications on R (with respect to the relationdefined in Definition 3.5).Proof. Let ξ : Ω( R ) −→ Ξ( R ) be the canonical surjection. By Lemma 8.1, ( σ, s ) isa homothetic datum on R if and only if ξ ( σ ) is an idempotent in Ξ( R ). Let σ, τ ∈ Ω( R ) = Π( R ) such that both ξ ( σ ) and ξ ( τ ) are equivalent idempotents. This meansthat there exist s, t, v ∈ R and a ring automorphism Φ : R −→ R such that(a) σ − σ = ¯ s ,(b) τ − τ = ¯ t ,(c) τ = Φ ∗ ( σ − ¯ v ).The claim (c) follows from the facts that ξ ( τ ) = Φ ⋄ ( ξ ( σ )), where Φ ⋄ is the ring ho-momorphism given by the diagram (3.10), and that Φ ∗ is an automorphism of Ω( R )mapping inner bimultiplications into inner ones. Therefore,¯ t = τ − τ = Φ ∗ (( σ − ¯ v ) − σ + ¯ v )= Φ ∗ ( σ − ¯ vσ − σ ¯ v + ¯ v − σ + ¯ v )= Φ ∗ (¯ s − ¯ vσ − σ ¯ v + ¯ v + ¯ v ) = Φ( s − vσ − σv + v + v ) . The last equality follows by the combination of (3.7) and (3.9). By Lemma 8.1, t = Φ( s − vσ − σv + v + v ) , and in view of Lemma 3.8, T ( σ, s ) ∼ = T ( τ, t ) as required.In the converse direction, in view of Lemma 3.9, if T ( σ, s ) ∼ = T ( τ, t ) then here exista ring automorphism Φ of R and an element v of R such that τ = Φ ∗ ( σ − ¯ v ) = Φ ∗ ( σ ) − Φ ∗ (¯ v ) = Φ ∗ ( σ ) − Φ( v ) , and hence ξ ( τ ) = Φ ⋄ ( ξ ( σ )), so that the corresponding idempotents in Ξ( R ) are equiv-alent in the sense of Definition 3.5. (cid:3) Theorem 8.2 has a quite surprising consequence for one-sided maximal ideals insimple rings with identity.
Theorem 8.3.
Let I be a maximal right (resp. left) ideal in a simple ring with identity R . Up to isomorphism there are exactly two trusses T on the heap H( I ) such that I = R( T ; 0) :(a) T( I ) , i.e. the truss with the same multiplication as that in I ,(b) the truss with multiplication given by the formula I × I −→ I, ( a, b ) ab + a + b. Proof.
The proof of this theorem relies on a connection between extensions of ringsto the ring of bimultiplications and maximal essential ring extensions introduced byBeidar [Be85], [Be93]. A ring ME( I ) is called a maximal essential extension of a ring I if I is an essential ideal in ME( I ), and if S is any ring that contains I as an essentialideal, then there exists a ring homomorphism ψ : S −→ ME( I ) such that ψ ( x ) = x ,for all x ∈ I . Beidar proves that if I is any right ideal of a ring R with identity suchthat RI = R , then ME( I ) = Id( I ) := { a ∈ R | aI ⊆ I } , (8.1)the idealiser of I in R , i.e. the largest subring of R containing I as an ideal. Inparticular, the equality (8.1) holds for any right ideal in a simple ring R with identity.The notion of an idealiser was introduced by Ore [Or32] and thoroughly studied byRobson in [Ro72]. In particular in [Ro72, Proposition 1.1] Robson proves an extensionof [Fi35, Satz 1], thus establishing an isomorphism of ringsId( I ) /I ∼ = End R ( R/I ) , a + I [ r + I ar + I ] . On the other hand a theorem of Flanigan [Fl78] (see [An11] for an elementary proof)states that a ring I admits a maximal essential extension if and only if it has a trivialannihilator. In that case I may be identified with the essential ideal I of Ω( I ). If S isany ring that contains I as an essential ideal, then the map ψ : S −→ Ω( I ) , s [¯ s : ( a sa, a as )] , is a ring homomorphism such that ψ ( a ) = ¯ a , for a ∈ I . Therefore ME( I ) ∼ = Ω( I ).If I is a right ideal in a simple ring R , then putting all this information together weobtain the following chain of isomorphisms of ringsΞ( I ) = Ω( I ) / ¯ I ∼ = ME( I ) /I ∼ = Id( I ) /I ∼ = End R ( R/I ) . If, furthermore, I is a maximal right ideal, then R/I is a simple right R -module,hence, by the Schur Lemma, End R ( R/I ) is a division ring, and so is the ring of outerbimultiplications Ξ( I ). Therefore, I and id + I are the only idempotents in Ξ( I ), andby Theorem 8.2, there are precisely two isomorphism classes of homothetic trusses on I . The corresponding rings (with the zero of I as the neutral element of the additivegroup) come out as stated in (a) and (b).The left ideal case is treated in an analogous way. (cid:3) We conclude this section with an extensive example which provides one with thefull classification (up to isomorphism) of trusses that can be constructed on the heapdetermined by the abelian group Z p ⊕ Z p , where p is a prime number. DEAL RING EXTENSIONS AND TRUSSES 37
Example 8.4.
It is well-known that there are eight isomorphism classes of rings builton the abelian group Z p ⊕ Z p . These are:(i) the field of p -elements, F p ;(ii) the product ring Z p × Z p ;(iii) the dual numbers ring Z p [ x ] / ( x );(iv) the zero ring Z p × Z p ;(v) the row matrix ring (cid:18) Z p Z p (cid:19) ;(vi) the column matrix ring (cid:18) Z p Z p (cid:19) ;(vii) the half-zero ring Z p × Z p ;(viii) the quotient ring x Z p [ x ] /x Z p [ x ].Rings (i)–(iii) have identity so by Proposition 3.12 each one of them admits exactlyone isomorphism class of homothetic trusses. The full classification of homothetic ringson the zero ring in (iv) can be obtained from Theorem 7.5. Following this theoremwe obtain ten such types of rings (the full list is given in the table below). Rings(v) and (vi) are maximal right respectively left ideals in a simple ring with identity,and so each one of them will admit exactly two non-isomorphic homothetic trusses byTheorem 8.3. The ring (vii) is the product of a ring with identity and the zero ring onan indecomposable group, hence the classification can be obtained by Proposition 3.13and Example 7.6, and there are four non-isomorphic trusses in this case. Thus itremains only to study the final case (viiii).The ring x Z p [ x ] /x Z p [ x ] is a commutative Z p -algebra with a basis x, x subject tothe relation x = 0. Let σ be a double homothetism and suppose that σx = ax + bx & xσ = cx + dx . Then the module properties imply that σx = ax and x σ = cx , and ( xσ ) x = x ( σx )implies that a = c . Now, the condition ( σx ) σ = σ ( xσ ) is automatically satisfied, andhence any homothetism is of the above form with a = c . We thus obtain σ x = a x + 2 abx & xσ = a x + 2 adx . Let s = s x + s x . Then the condition σ = σ + ¯ s yields, a = a & s = 2 ab − b = 2 ad − d. Since Z p is a field, b = d and there are only the following solutions to these equations: a = 0, s = − b and a = 1, s = b . In the first case σ is the inner homothetism σ = bx ,and so the corresponding truss has the same multiplication as R . In the other case,one easily finds that the corresponding truss is isomorphic to T(id ,
0) by setting Φ = idand v = bx − s x . These two trusses are not isomorphic by Proposition 3.12. since x Z p [ x ] /x Z p [ x ] has no identity.Since every truss on a given abelian heap is a homothetic truss on the ring on thecorresponding abelian group, we have obtained a full classification of non-isomorphictrusses on the heap H( Z p ⊕ Z p ). There are 23 such (classes of) trusses, which we listin the following table. Product ⋄ in the truss T( σ, s ),Ring R on Z p ⊕ Z p for all a = ( a , a ) , b = ( b , b ) ∈ R F p a ⋄ b = ab Z p × Z p a ⋄ b = ab Z p [ x ] / ( x ) a ⋄ b = aba ⋄ b = 0; a ⋄ b = b ; a ⋄ b = a ; a ⋄ b = a + b ; Z p × Z p a ⋄ b = ( a , a ⋄ b = (0 , b ); a ⋄ b = ( a + b , b ); a ⋄ b = ( a + b , a ); a ⋄ b = ( a + b , a ⋄ b = ( a , b ) (cid:18) Z p Z p (cid:19) a ⋄ b = ab ; a ⋄ b = ab + a + b (cid:18) Z p Z p (cid:19) a ⋄ b = ab ; a ⋄ b = ab + a + b Z p × Z p a ⋄ b = ( a a , b ); a ⋄ b = ( a a , b ); a ⋄ b = ( a a , a ⋄ b = ( a a , b + b ) x Z p [ x ] /x Z p [ x ] a ⋄ b = ab ; a ⋄ b = ab + a + b Coda Conclusions and outlook
The main aim of this work was to place a novel theory of trusses on the more familiarground of classical ring theory. The results presented here show that trusses can beviewed as a different way of dealing with ring extensions, more precisely those that arisefrom Redei’s double homothetisms or ideal extensions of rings by the ring of integers.In short one can make the following heuristic statement: up to translations trussesare equivalence classes of ring extensions by integers . Despite this closenessof trusses and ideal ring extensions the results of the paper show that trusses cannotbe reduced to rings. Our final example, which demonstrates that there are 23 different
DEAL RING EXTENSIONS AND TRUSSES 39 isomorphism classes of trusses on the abelian group Z p × Z p , as opposed to only 8 ringsclasses on the same group, might serve as a justification for this claim.From a universal algebra point of view trusses are not as complicated algebraic sys-tems as rings, let alone, ring extensions, hence, in our opinion should hopefully provideone with quite an effective way of describing such extensions. On the one hand theresults of this paper point to applications of trusses to ring theory and homologicalalgebra, while on the other the vast existing knowledge about ring extensions shouldfeed into the theory of trusses. For example, studying all possible (and, specifically,non-trivial) ring structures supported on a given abelian group or characterising groupsby types of rings which they support are long-standing problems in algebra (see e.g.[NW17]). This can now be translated into the classification problem of trusses. Clas-sification of ring extensions, normally undertaken by homological methods, can bereplaced by classification of homothetic trusses on a given ring. Since all extensions ofrings can be understood as arising from families of permutable or amicable homoth-etisms, developing the connections described in this paper to families of homothetismsmight produce new techniques for studying more general classes of extensions. Trussesarose as an attempt to understand connections between braces and rings. Once develop-ments presented here are extended to near-rings and their extensions, novel methods ofclassifying and constructing braces and thus solutions of the set-theoretic Yang-Baxterequation might be obtained. One of the key obstacles to study categories of modulesover trusses, such as those encountered in the definition of projectivity [BRS20], arisefrom the fact that modules over trusses are enriched over the category of abelian heaps,which is not an abelian category. Employing the ring theoretic language of extensionsand thus working over an abelian category, might lead to better understanding whatmodules over a truss really are or how they can be defined in the most effective way.In short, we believe that the structural simplicity of trusses and the richness of theirconnections with other, well-understood, algebraic systems, while allowing one to “playgracefully with ideas” (Oscar Wilde, De Profundis ), will lead to an enhancement of thealgebraic landscape.
Acknowledgements
The research of the first two authors is partially supported by the National ScienceCentre, Poland, grant no. 2019/35/B/ST1/01115. We would like to thank KarolPryszczepko and all other members of the Department of Algebra at the University ofBia lystok for interesting discussions.
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DEAL RING EXTENSIONS AND TRUSSES 41
Faculty of Mathematics, University of Bia lystok, K. Cio lkowskiego 1M, 15-245 Bia- lystok, Poland
Email address : [email protected] Department of Mathematics, Swansea University, Swansea University Bay Campus,Fabian Way, Swansea, Swansea SA1 8EN, U.K.Faculty of Mathematics, University of Bia lystok, K. Cio lkowskiego 1M, 15-245 Bia- lystok, Poland
Email address : [email protected]
Department of Mathematics, Swansea University, Swansea University Bay Campus,Fabian Way, Swansea, Swansea SA1 8EN, U.K.
URL : https://sites.google.com/view/bernardrybolowicz/ Email address ::