The Baer-Kaplansky theorem for all abelian groups and modules
aa r X i v : . [ m a t h . G R ] J a n THE BAER-KAPLANSKY THEOREM FOR ALL ABELIAN GROUPSAND MODULES
SIMION BREAZ AND TOMASZ BRZEZI ´NSKI
Abstract.
It is shown that the Baer-Kaplansky theorem can be extended to allabelian groups provided that the rings of endomorphisms of groups are replaced bytrusses of endomorphisms of corresponding heaps. That is, every abelian group isdetermined up to isomorphism by its endomorphism truss and every isomorphismbetween two endomorphism trusses associated to some abelian groups G and H isinduced by an isomorphism between G and H and an element from H . This corre-spondence is then extended to all modules over a ring by considering heaps of modules.It is proved that the truss of endomorphisms of a heap associated to a module M determines M as a module over its endomorphism ring. Introduction
The Baer-Kaplansky Theorem, [14, Theorem 16.2.5], states that, for every isomor-phism Φ : End( G ) −→ End( H ) between the endomorphism rings of torsion abeliangroups G and H there exists an isomorphism ϕ : G −→ H such that Φ : α ϕαϕ − (here and throughout this note the composition of mappings is denoted by juxtaposi-tion of symbols). A similar result was known for endomorphisms of vector spaces. Theexistence of an analogous statement for other kinds of modules (or abelian groups) wasinvestigated by many authors, e.g. in [12], [15], [18], and [19]. There are situations inwhich there exist isomorphisms between endomorphism rings that are not induced inthe above mentioned way. Such an example is described in [15, p. 486], and some otherdetailed studies are presented in [12] and [19]. On the other hand, there are exampleswhich show that, in general, in order to obtain a Baer-Kaplansky Theorem, one needsto restrict to some reasonable classes of objects, see e.g. [14, Example 9.2.3]. From adifferent perspective, it was proven in [2] that in the case of modules over principalideal domains every module is determined up to isomorphism by the endomorphismring of a convenient module. In particular, two abelian groups G and H are isomorphicif and only if the rings End( Z ⊕ G ) and End( Z ⊕ H ) are isomorphic. However, if G contains a direct summand isomorphic to Z then not every ring isomorphism betweenEnd( Z ⊕ G ) and End( Z ⊕ H ) is induced in a natural way by an isomorphism between G and H .A natural question that emerges from this discussion is this: can one associate someendomorphism structures to abelian groups such that all isomorphisms between two Mathematics Subject Classification.
Primary: 20K30; Secondary: 16Y99.
Key words and phrases. abelian group, heap, endomorphism truss.This research of T. Brzezi´nski is partially supported by the National Science Centre, Poland, grantno. 2019/35/B/ST1/01115. such endomorphism structures are induced in a natural way by isomorphisms betweenthe corresponding abelian groups?In this note we present an answer to this question that makes use of an observationmade by Pr¨ufer [16] that the structure of an abelian group ( G, +) can be fully encodedby a set G with a ternary operation modelled on [ a, b, c ] = a − b + c (this observationwas extended to general groups by Baer [1]). A set G together with such a ternaryoperation is called a heap , and a heap is abelian if [ a, b, c ] = [ c, b, a ], for all a, b, c ∈ G ;the precise definition is recalled in the following section. By fixing the middle entryin the ternary operation, a (abelian) group can be associated to a (abelian) heapuniquely up to isomorphism. In this way we obtain a bijective correspondence betweenthe isomorphism classes of (abelian) groups and the isomorphism classes of (abelian)heaps. Details about these correspondences are given e.g. in [5], [9], and [13]. The set E ( G ) of all endomorphisms of an abelian heap is a heap with the ternary operationdefined pointwise. Being an endomorphism object it has a natural monoid structuregiven by composition. The composition distributes over the ternary heap operationthus making E ( G ) a (unital) truss , a notion proposed in [4] as an algebraic structureencapsulating both rings and (skew) braces introduced in [17], [8], [11] to capture thenature of solutions to the set theoretic Yang-Baxter equation [10].The endomorphism truss E ( G ) of an abelian group G carries more information thanthe endomorphism ring End( G ) (just as the holomorph of a group carries more infor-mation than the group of its automorphisms). For example E ( G ) includes all constantmappings. In fact E ( G ) can be realised as a semi-direct product G ⋊ End( G ), see[5, Proposition 3.44]. It seems quite reasonable to expect that E ( G ) provides a rightenvironment for the Baer-Kaplansky theorem for all abelian groups. In this note weshow that this is indeed the case, and the first main result, Theorem 2.2, establishesthe Baer-Kaplansky type correspondence between the isomorphisms of abelian groups( G, +) and ( H, +), and isomorphisms of their endomorphism trusses E ( G ) and E ( H ).In the second main result, Theorem 3.3 we extend the Baer-Kaplansky correspondenceto all modules M over a ring R . To achieve this we associate to each module over aring a family of modules, which we term the heap of modules and we define homo-morphisms of heaps of modules as maps that respect all these module structures ina specific way. Endomorphisms of heaps of modules form trusses, and these trussesfor two modules (over possibly different rings) are isomorphic if and only if the mod-ules are equivalent as modules over their own endomorphism rings. The equivalenceof modules over different rings is defined in a natural way as an isomorphism in thecategory consisting of pairs of rings and (left) modules over these rings ( R, M ), withmorphisms from (
R, M ) to (
S, N ) given as pairs consisting of a ring homomorphism ̺ : R −→ S and a homomorphism of modules µ : M −→ N over R , with N viewed asthe R -module via ̺ .2. Morphisms between endomorphism trusses A heap is a set G together with a ternary operation [ − , − , − ] on G that is associativeand satisfies the Mal’cev identities, that is, for all a, b, c, d, e ∈ G ,[[ a, b, c ] , d, e ] = [ a, b, [ c, d, e ]] and [ a, a, b ] = b = [ b, a, a ] . HE BAER-KAPLANSKY THEOREM FOR ALL ABELIAN GROUPS AND MODULES 3
The heap ( G, [ − , − , − ]) is abelian if [ a, b, c ] = [ c, b, a ] for all a, b, c ∈ G .A heap structure on a set G induces a group structure on G , and this group structureis unique up to isomorphism. More precisely, if ( G, [ − , − , − ]) is a heap and b ∈ G thenthe binary operation + b defined by a + b c = [ a, b, c ] , for all a, b, c ∈ G, equips G with a group structure. The group ( G, + b ) is referred to as a retract of( G, [ − , − , − ]) at b . The operation + b is commutative provided G is an abelian heap.Moreover, if b ′ ∈ G then the groups ( G, + b ) and ( G, + b ′ ) are isomorphic by the map a [ a, b, b ′ ].Conversely, if ( G, +) is a group then G together with the ternary operation[ a, b, c ] = a − b + c, for all a, b, c ∈ G, is a heap, which is abelian if and only if ( G, +) is abelian. In this case, for every b ∈ G ,the induced group structure ( G, + b ) is isomorphic to ( G, +) since + = + . Unlike theassignment of a retract to a heap, this assignment of the heap to a group is functorial,that is, it defines a functor from the category of (abelian) groups to that of (abelian)heaps.If ( G, [ − , − , − ]) and ( H, [ − , − , − ]) are heaps, a mapping α : G −→ H is a heapmorphism if α ([ a, b, c ]) = [ α ( a ) , α ( b ) , α ( c )], for all a, b, c ∈ G . The set of all heapmorphisms from G to H is denoted by Heap( G, H ). If G and H are abelian heaps,then Heap( G, H ) is a heap with the pointwise defined heap operation [ α, β, γ ]( a ) =[ α ( a ) , β ( a ) , γ ( a )]. We note that α : G −→ H is a heap morphism if and only if α : ( G, + b ) −→ ( H, + α ( b ) ) is a group morphism for every b ∈ G . If G and H are groups,a map α : G −→ H is a heap morphism if it is a morphism between the induced heapstructures on G and H . Every group morphism is a heap morphism, but this property isnot reciprocated. In a standard way, a heap morphism with equal domain and codomainis called a heap endomorphism . The set of all heap endomorphisms of an abelian group G is denoted by E ( G ). Clearly, E ( G ) is a monoid with the composition of maps as amultiplicative operation. However, the multiplicative operation (composition) does notdistribute over the pointwise defined addition. On the other hand, it does distributeover the pointwise defined ternary heap operation [ α, β, γ ]( a ) = α ( a ) − β ( a ) + γ ( a ),that is, δ [ α, β, γ ] = [ δα, δβ, δγ ] and [ α, β, γ ] δ = [ αδ, βδ, γδ ] , for all α, β, γ, δ ∈ E ( G ). An abelian heap together with a semigroup operation thatdistributes over the heap operation in the above sense is called a truss . A morphismof trusses is a mapping that is both a homomorphism of heaps and semigroups. Thusthe algebraic system ( E ( G ) , [ − , − , − ] , · ) is a truss.If a ∈ G , then we will denote by b a the constant map b a : G −→ G, b a. Thanks to the fact that the heap operation is idempotent, every b a is an element on E ( G ). Clearly, the set b G of all constant maps is closed under the heap operation andthe composition, hence it forms a sub-truss of E ( G ). The map b : G −→ b G is anisomorphism of heaps. The multiplication in E ( G ) transferred back to G equips the SIMION BREAZ AND TOMASZ BRZEZI ´NSKI heap ( G, [ − , − , − ]) with the lopsided truss structure ab = a , for all a, b ∈ G (of noimportance in this note, though).The proof of the following characterization for heap morphisms is a simple exercise. Lemma 2.1.
Let ( G, +) and ( H, +) be abelian groups. (i) A map ϕ : G −→ H is a heap morphism (resp. isomorphism) if and onlyif there exists a group morphism (resp. isomorphism) e ϕ : G −→ H and anelement h ∈ H such that ϕ ( ) = e ϕ ( ) + h . In this case, e ϕ ( ) = ϕ ( ) − ϕ (0) and h = ϕ (0) are uniquely determined by ϕ . (ii) A heap morphism ϕ : G −→ H is constant if and only if ϕ ˆ a = ϕ , for all ˆ a ∈ b G .In this case ϕα = ϕ , for all α ∈ E ( G ) . (iii) For all a ∈ G , ϕ b a = d ϕ ( a ) . The first main result of this note is contained in the following theorem.
Theorem 2.2 (The Baer-Kaplansky theorem for abelian groups) . Two abelian groups ( G, +) and ( H, +) are isomorphic if and only if their endomorphism trusses E ( G ) and E ( H ) are isomorphic.Furthermore, for every isomorphism Φ : E ( G ) −→ E ( H ) of trusses there exists aunique heap isomorphism ϕ : G −→ H such that Φ( α ) = ϕαϕ − for all α ∈ E ( G ) .Proof. For all a ∈ G and b ∈ H ,Φ( b a ) b b = Φ( b a Φ − ( b b )) = Φ( b a ) . It follows that Φ( b a ) ∈ b H . Therefore, one can define functions ϕ : G −→ H and ψ : H −→ G , by setting d ϕ ( a ) = Φ( b a ) , d ψ ( b ) = Φ − ( b b ) , for all a ∈ G and b ∈ H . Note that, since Φ( b a ) is a constant function, ϕ ( a ) = Φ( b a )( b ),for all b ∈ H (in particular for b = 0). Since both Φ and Φ − are heap morphisms, ϕ and ψ are heap morphisms too. Moreover, ϕψ = 1 H since \ ϕψ ( b ) = Φ( d ψ ( b )) = Φ(Φ − ( b b )) = b b, for all b ∈ H . In an analogous way one proves that ψϕ = 1 G , hence ψ = ϕ − and ϕ isan isomorphism of heaps. In view of Lemma 2.1 the required isomorphism of abeliangroups is thus obtained as a ϕ ( a ) − ϕ (0).The assignment Φ ϕ just constructed can be understood as the maping Θ,defined by (Φ : E ( G ) −→ E ( H )) Θ (Φ( b )(0) : G −→ H ) , from the set of truss isomorphisms E ( G ) −→ E ( H ) to the set of heap isomorphisms G −→ H . The inverse map Υ is defined by Υ( ϕ )( α ) = ϕαϕ − , for all heap isomor-phisms ϕ : G −→ H and all α ∈ E ( G ). Clearly, Υ( ϕ ) is an isomorphism of trusses.To check that Θ and Υ are mutual inverses, take any heap isomorphism ϕ : G −→ H and a ∈ G , and compute \ ΘΥ( ϕ )( a ) = (Υ( ϕ ))( b a ) = ϕ b aϕ − = ϕ b a = d ϕ ( a ) , HE BAER-KAPLANSKY THEOREM FOR ALL ABELIAN GROUPS AND MODULES 5 where the last two equalities follow by Lemma 2.1. Therefore, ΘΥ( ϕ ) = ϕ , for all heapisomorphisms ϕ : G −→ H .In the converse direction, for all truss isomorphisms Φ : E ( G ) −→ E ( H ), and α ∈ E ( G ), both Θ(Φ) α and Φ( α )Θ(Φ) are heap morphisms from G onto H . Moreover,for all a ∈ G , \ Θ(Φ) α ( a ) = Φ( d α ( a )) = Φ( α b a ) = Φ( α )Φ( b a ) = Φ( α ) \ Θ(Φ)( a ) . It follows that Θ(Φ) α = Φ( α )Θ(Φ), whenceΥΘ(Φ)( α ) = Θ(Φ) α Θ(Φ) − = Φ( α ) , for all α ∈ E ( G ). This completes the proof. (cid:3) Remark . In the first part of the proof of Theorem 2.2 it was shown that Φ( b G ) ⊆ b H .Let us mention that this conclusion is valid for all surjective semigroup morphisms( E ( G ) , · ) −→ ( E ( H ) , · ). This can be also obtained by using the equality b G = { α ∈ E ( G ) | αβ = α for all β ∈ E ( G ) } , meaning that the set of constant maps G −→ G coincides with the set of left absorbingelements in the semigroup ( E ( G ) , · ).Since b G ∼ = G as heaps and every truss isomorphism E ( G ) −→ E ( H ) restricts to thetruss isomorphism b G −→ b H , it induces a heap isomorphism G −→ H . Remark . In view of the correspondence between abelian groups and heaps, inparticular, since the transformation of a heap to any of its retracts and then back tothe heap yields identity, one can reformulate Theorem 2.2 and prove it entirely in theheap phraseology. Specifically, the mapping Θ in the proof of Theorem 2.2 establishes abijective correspondence between isomorphisms of all abelian heaps and isomorphismsbetween their corresponding endomorphism trusses.In fact, all truss morphisms E ( G ) −→ E ( H ) are inner in some sense. Proposition 2.5.
Let ( G, +) , ( H, +) be abelian groups and let Φ : E ( G ) −→ E ( H ) bea truss morphism. Set ε Φ = Φ( b − Φ( b and e Φ = Φ( b , and define the following subset of the heap Heap(
G, H ) of all heap morphisms from G to H , Ξ Φ := { ξ ∈ Heap(
G, H ) | Φ( α ) ξ = ξα, for all α ∈ E ( G ) } . (a) The map ε Φ is an idempotent endomorphism of the abelian group ( H, +) and ε Φ ( e Φ ) = 0 . (b) The set Ξ Φ is not empty. (c) For all ξ ∈ Ξ Φ , ξ (0) ∈ e Φ + Im( ε Φ ) . (d) The set Ξ Φ is a sub-heap of Heap(
G, H ) , and it is isomorphic to the sub-heap e Φ + Im( ε Φ ) of H .Proof. (a) From Φ( b
0) = Φ( b b
0) it follows that ε Φ + e Φ = ( ε Φ + e Φ )( ε Φ + e Φ ) = ε Φ ε Φ + ε Φ ( e Φ ) + e Φ . Using Lemma 2.1, we obtain that ε Φ is idempotent and ε Φ ( e Φ ) =0. SIMION BREAZ AND TOMASZ BRZEZI ´NSKI (b) Let b ∈ H and define ξ = ξ b : G −→ H, a Φ( b a )( b ) . Then, for every α ∈ E ( G ),Φ( α ) ξ b ( a ) = Φ( α )Φ( b a )( b ) = Φ( α b a )( b ) = Φ( d α ( a ))( b ) = ξ b ( α ( a )) , hence ξ b ∈ Ξ Φ .(c) For any ξ ∈ Ξ Φ , ξ (0) = ξ ( b b ξ (0) ∈ Im( ε Φ ) + e Φ , where the second equality follows by the definition of Ξ Φ .(d) The first statement is obvious. Since Im( ε Φ ) + e Φ is a coset it is a sub-heap of H by [9, Theorem 1]. Using the same notation as in the proof of (b), we consider themap ϑ : Im( ε Φ ) + e Φ −→ Ξ Φ , c ξ c . It is easy to see that this is a morphism of heaps, so it remains only to prove that it isbijective.Take any ξ ∈ Ξ Φ . Then, for all a ∈ G , ξ ( a ) = ξ ( b a (0)) = Φ( b a ) ξ (0) = ξ ξ (0) ( a ) . Hence ξ = ξ ξ (0) , and the map ϑ is surjective.Moreover, we observe that for every c ∈ Im( ε Φ ) + e Φ , ξ c (0) = Φ( b c ) = ε Φ ( c ) + e Φ = c + e Φ , since ε Φ is an idempotent group endomorphism and ε Φ ( e Φ ) = 0. Therefore, if c, d ∈ Im( ε Φ ) + e Φ and c = d then ξ c = ξ d , and the proof is completed. (cid:3) Corollary 2.6.
Let ( G, +) and ( H, +) be abelian groups, and let Φ : E ( G ) −→ E ( H ) be a truss morphism. Suppose that there exists a ∈ G such that Φ( b a ) is a constantmorphism. Then there exists a unique morphism of heaps ξ : G −→ H such that Φ( α ) ξ = ξα, for all α ∈ E ( G ) . Proof.
Let us observe that Φ( b
0) = Φ( b b a ) is a constant morphism. It follows that ε Φ = 0. As a consequence of Proposition 2.5, there is a unique heap morphism ξ suchthat Φ( α ) ξ = ξα for all α ∈ E ( G ). (cid:3) Isomorphisms of modules over endomorphism rings
Next, we would like to extend the Baer-Kaplansky correspondence to modules overa ring R . Na¨ıvely one could try to establish the correspondence between R -moduleisomorphisms and endomorphism trusses of R -modules. One could try to follow thestrategy of Theorem 2.2, that is, view the additive structure of an R -module M asa heap and then study the R -linear heap endomorphisms of M , that is, heap endo-morphisms that commute with the R -action. There are at least two difficulties withimplementing this strategy. The first and rather technical problem is that the con-stant heap endomorphism is not R -linear in general, hence the arguments of the proofof Theorem 2.2 may only be carried over in this limited way. The second and more HE BAER-KAPLANSKY THEOREM FOR ALL ABELIAN GROUPS AND MODULES 7 fundamental point is that, as observed in [6, Lemma 4.5], an R -linear heap endomor-phism of M necessarily maps the zero of M into itself. As a consequence the set ofall R -linear endomorphisms of the heap M coincides with the set of endomorphismsof M over the ring R . The zero map ˆ0 is the absorber of the latter understood as atruss with respect of composition, that is, α ˆ0 = ˆ0 α = ˆ0 (note that this is not the casein E ( M ) but only in the R -linear part of E ( M )). Since homomorphisms of trussespreserve absorbers and the retract of a truss at an absorber is a ring, isomorphisms oftrusses of endomorphisms of modules over a ring coincide with isomorphisms of corre-sponding endomorphism rings. And the class of these is too restrictive to capture allisomorphisms between modules.Finally, let us stress that preceding remarks do not contradict the validity of Theo-rem 2.2. Although any abelian group G is a Z -module, by looking at E ( G ) and treating G as a heap we depart from viewing it not only as a module over the ring Z but alsoas a module over the associated truss. This provides one with the required flexibilityof the structure to capture all isomorphisms of abelian groups.In turns out that to implement the Baer-Kaplansky theorem for modules over ringa new concept of the truss of R -linear heap endomorphisms is needed.Recall from [5] that, given a truss T , an abelian heap M together with the asso-ciative action · of the multiplicative semigroup of T that distributes over the ternaryoperations, that is, for all t, t , t , t ∈ T and m, m , m , m ∈ M ,[ t , t , t ] · m = [ t · m, t · m, t · m ] , t · [ m , m , m ] = [ t · m , t · m , t · m ] , is called a left T -module . Any element e ∈ M induces a new T -module structure on M with the action t e · m = [ t · m, t · e, e ] , for all t ∈ T and m ∈ M .
Every ring R can be viewed as a truss with the same multiplication as that in R and with the (abelian group) heap structure [ r, s, t ] = r − s + t . A left module ( M, +)over a ring R is a left module over the truss R : M is viewed as a heap in a naturalway and the R -action · is unchanged (we will denote it by juxtaposition). Note thatin this case the scalar multiplication e · is exactly the multiplication induced from the R -action on M by imposing the condition that the isomorphism of abelian groups( M, +) −→ ( M, + e ), x [ x, , e ], is an isomorphism of R -modules. In order to seethis, write every element m ∈ M as m = [[ m, e, , , e ]. The above condition is thenequivalent to r e · m = [ r [ m, e, , , e ] = [ rm, re, e ] for all m ∈ M. Since 0 · m = 0, the induced action · is equal to the original action · .Given a left module M over a ring R , the family H ( M ) of all R -modules ( M, + e )whose scalar multiplications are e · is called the heap of the module M . If M, N are R -modules, a morphism of heaps of modules , ϕ : H ( M ) −→ H ( N ), is a map ϕ : M −→ N that for every e ∈ M is a homomorphism of R -modules ϕ : ( M, + e ) −→ ( N, + ϕ ( e ) ). Wedenote by H R ( M, N ) the set of all morphisms of heaps of modules H ( M ) −→ H ( N ).The proof of the following lemma is straightforward. SIMION BREAZ AND TOMASZ BRZEZI ´NSKI
Lemma 3.1.
For a ring R and left R -modules M and N , H R ( M, N ) := (cid:8) ϕ ∈ Heap(
M, N ) | ϕ ( rm ) = rϕ ( m ) − rϕ (0) + ϕ (0)= [ rϕ ( m ) , rϕ (0) , ϕ (0)] , ∀ r ∈ R, m ∈ M (cid:9) . Moreover, (i)
A heap homomorphism (resp. isomorphism) ϕ : M −→ N is an element of H R ( M, N ) if and only if the map e ϕ : M −→ M, m ϕ ( m ) − ϕ (0) , is a homomorphism (resp. isomorphism) of R -modules. (ii) Hom R ( M, N ) ⊆ H R ( M, N ) . (iii) The set of endomorphisms of the heap of M , E R ( M ) := H R ( M, M ) , is a sub-truss of E ( M ) .Remark . Another way of interpreting the set H R ( M, N ) is provided by observingthat the correspondence described in the statements of Lemma 3.1 can be lifted to thebijection H R ( M, N ) ∼ = −→ N × Hom R ( M, N ) , ϕ (cid:0) ϕ (0) , e ϕ (cid:1) . The inverse is given by N × Hom R ( M, N ) −→ H R ( M, N ) , (cid:0) n, e ϕ (cid:1) (cid:2) ϕ : m n + e ϕ ( m ) (cid:3) . The heap structure induced by this isomorphism is that of the product of heaps, thatis, (cid:2) ( n , e ϕ ) , ( n , e ϕ ) , ( n , e ϕ ) (cid:3) = (cid:0) n − n + n , e ϕ − e ϕ + e ϕ (cid:1) , for all n , n , n ∈ N and e ϕ , e ϕ , e ϕ ∈ Hom R ( M, N ).In the following result we will prove that E R ( M ) determines a left R -module M asa module over its endomorphism ring End R ( M ) with the action given by evaluation.If R and S are rings, M is a left R -module, and N is a left S module, then we say thatthe modules M and N are equivalent if and only if there exists a group isomorphism µ : M −→ N and a ring isomorphism ̺ : R −→ S such that µ ( rm ) = ̺ ( r ) µ ( m ) for all m ∈ M and r ∈ R . Theorem 3.3 (The Baer-Kaplansky theorem for modules) . Let R and S be rings, andlet M be a left R -module and N be a left S -module. The trusses E R ( M ) and E S ( N ) areisomorphic if and only if M and N are equivalent as modules over their endomorphismrings.Proof. If M and N are equivalent as modules over their endomorphism rings, there existan additive map µ : M −→ N and a ring isomorphism ̺ : End R ( M ) −→ End S ( N )such that µ ( um ) = ̺ ( u ) µ ( m ), for all m ∈ M and u ∈ End R ( M ), or, equivalently, µu = ̺ ( u ) µ . We claim thatΦ : E R ( M ) −→ E S ( N ) , α ̺ ( α − α (0)) + µ ( α (0)) = ̺ ( e α ) + µ ( α (0)) , is an isomorphism of trusses. Indeed, first noting that Φ( α )(0) = µ ( α (0)), and usingthat ̺ maps endomorphisms of M into endomorphisms of N one easily finds that HE BAER-KAPLANSKY THEOREM FOR ALL ABELIAN GROUPS AND MODULES 9 Φ( α ) ∈ E S ( N ). To prove that Φ respects multiplications, it is enough to observe that,for all α, β ∈ E R ( M ), e α e β = f αβ, since, for all m ∈ M , e α e β ( m ) = α ( β ( m ) − β (0)) − α (0)= α ( β ( m )) − α ( β (0)) + α (0) − α (0) = f αβ ( m ) , where we have used that α is a heap endomorphism. Therefore, using the definition ofthe equivalence of modules we find,Φ( αβ ) = ̺ ( f αβ ) + µ ( αβ (0)) = ̺ ( e α ) ̺ ( e β ) + µ ( e αβ (0) + α (0))= ̺ ( e α ) ̺ ( e β ) + ̺ ( e α ) µ ( β (0)) + µ ( α (0))= Φ( α ) (cid:16) ̺ ( e β ) + µ ( β (0)) (cid:17) = Φ( α )Φ( β ) , as required.Conversely, let Φ : E R ( M ) −→ E S ( N ) be an isomorphism of trusses. As before,if m ∈ M , we denote by b m the constant map x m for all x ∈ M . Once it isnoted that, for all m ∈ M , b m ∈ E R ( M ) the arguments of the first part of the proof ofTheorem 2.2 may be repeated verbatim to associate a heap isomorphism ϕ : M −→ N corresponding to the truss isomorphism Φ : E R ( M ) −→ E S ( N ) by the formula [ ϕ ( m ) = Φ( b m ) , for all m ∈ M .
Define the isomorphism of abelian groups µ : M −→ N by µ = e ϕ , and set ̺ : End R ( M ) −→ E S ( N ) , u Φ( u ) − Φ( u )(0) = [Φ( u ) , \ Φ( u )(0) , b . Since ̺ ( u )(0) = 0, it follows that ̺ ( u ) ∈ End S ( N ) for all u ∈ End R ( M ). Furthermore,since Φ( u + v ) = Φ([ u, b , v ]) = [Φ( u ) , Φ( b , Φ( v )] = Φ( u ) + Φ( v ) − Φ( b , it can be proven that ̺ is additive by direct calculations.For all u ∈ End R ( M ) and m ∈ M we can compute, ̺ ( u ) µ ( m ) = [Φ( u )([ ϕ ( m ) , ϕ (0) , , Φ( u )(0) , u )( ϕ ( m )) , Φ( u )( ϕ (0)) , Φ( u )(0)] , Φ( u )(0) , u )( ϕ ( m )) , Φ( u )( ϕ (0)) , u )(Φ( b m )(0)) , Φ( u )(Φ( b , u b m )(0) , Φ( u b ,
0] = [Φ( [ u ( m ))(0) , Φ( b , \ ϕ ( u ( m ))(0) , d ϕ (0)(0) ,
0] = [ ϕ ( u ( m )) , ϕ (0) ,
0] = µu ( m ) . Hence ̺ ( u ) µ = µu . In particular, ̺ ( u ) = µuµ − , which immediately implies that ̺ isan injective multiplicative map.Let v ∈ End S ( N ) and α ∈ E R ( M ) such that Φ( α ) = v . Then ̺ ( e α ) = Φ( e α ) − Φ( e α )(0) = Φ([ α, d α (0) , b − Φ([ α, d α (0) , b α ) , Φ( d α (0)) , Φ( b − [Φ( α ) , Φ( d α (0)) , Φ( b α ) − Φ( α )(0) = v − v (0) = v, where we have used that Φ is a heap homomorphism and that it maps constant homo-morphisms to constant ones. Hence ̺ is surjective. Therefore, ̺ is a ring isomorphismas required. (cid:3) In general, the existence of an isomorphism of trusses E R ( M ) ∼ = E R ( N ) does notguarantee that the initial R -modules are isomorphic. Example . Let F be a field and R = F × F . If M = F × N = 0 × F , it is easyto check that E R ( M ) ∼ = E R ( N ), while there is no R -module isomorphism connecting M with N . Remark . As in the case of abelian groups, every truss isomorphism Φ : E R ( M ) −→ E S ( N ) is induced uniquely by a heap isomorphism ϕ ∈ H ( M, N ) defined in the proofof Theorem 3.3, via Φ( α ) = ϕαϕ − , for all α ∈ E R ( M ). However, not all heapisomorphisms ϕ : M −→ N induce an isomorphism of trusses E R ( M ) −→ E S ( N ) inthis way since in general ] Φ( α ) = ϕαϕ − − ϕαϕ − (0) need not be S -linear.For instance, let F = R in Example 3.4. Since √ √
3, and √ Q , there exists an additive isomorphism ϕ : R −→ R such that ϕ ( √
2) = √ ϕ ( √
3) = √
6, and ϕ ( √
6) = √
2. If α : R −→ R , α ( x ) = √ · x ,then ϕαϕ − ( √
2) = ϕ ( √ √
6) = ϕ (2 √
3) = 2 √
6, and ϕαϕ − ( √
6) = ϕ ( √ · √
3) = √ ϕαϕ − ( √ · √ = √ ϕαϕ − ( √ ϕαϕ − is not R -linear.4. Conclusions
In this note we have formulated and proven the Baer-Kaplansky theorem for allabelian groups and all modules over not necessarily commutative rings. It turns outthat such a formulation is possible provided we abandon the classical group and ringtheory point of view and adopt a more general albeit less standard perspective of heapsand trusses. This departure from a classical universe of groups and rings is not madeout of choice but out of necessity as it seems impossible to confine the informationabout isomorphism classes of abelian groups to a structure consisting of a set withtwo binary operations, one distributing over the other. Trusses that feature in theBaer-Kaplansky theorem may be interpreted as arising from a modification of thenotion of a group homomorphism; rather than dealing with group homomorphismsone needs to deal with homomorphisms of associated heaps instead. In a similar way,the Baer-Kaplansky theorem for modules relies on a modification of the notion of ahomomorphism of modules. The latter points to an exciting possibility of developinga new approach to module theory in which a single module over a ring is replaced bya heap of modules (as defined in Section 3). While propelled by the philosophy ofexploring fully ternary formulation of group axioms, this still goes beyond the studyof modules over trusses already undertaken in [5], [6] or [7] and constitutes the subjectof on-going investigations [3].
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