Axioms for Commutative Unital Rings elementarily Equivalent to Restricted Products of Connected Rings
aa r X i v : . [ m a t h . L O ] J u l AXIOMS FOR COMMUTATIVE UNITAL RINGSELEMENTARILY EQUIVALENT TO RESTRICTED PRODUCTSOF CONNECTED RINGS
JAMSHID DERAKHSHAN AND ANGUS MACINTYRE † Abstract.
We give axioms in the language of rings augmented by a 1-arypredicate symbol
F in ( x ) with intended interpretation in the Boolean algebraof idempotents as the ideal of finite elements, i.e. finite unions of atoms. Weprove that any commutative unital ring satisfying these axioms is elementarilyequivalent to a restricted product of connected rings. This is an extension ofthe results in [3] for products. While the results in [3] give a converse to theFeferman-Vaught theorem for products, our results prove the same for restrictedproducts. We give a complete set of axioms in the language of rings for the ringof adeles of a number field, uniformly in the number field. Introduction
This paper is a natural sequel to [3] and the main results and proofs are naturalextensions of those in [3]. In many cases we will simply refer to the material from[3][3] deals with the model theory of products of connected unital rings, and canbe construed as providing a partial converse to the Feferman-Vaught Theorem [9]in the special case of products Q i ∈ I R i , i ∈ I, where R i are connected commutativeunital rings and I an index set (Recall that a commutative ring R is connected if , are the only idempotents of R ). The converse concerns the issue of providingaxioms for rings elementarily equivalent to rings Q i ∈ I R i as above . The solution ofthis problem is given in [3] and, inter alia has applications to non-standard modelsof PA (first order Peano arithmetic) in [4].In this paper we start with rings Q i ∈ I R i , i ∈ I, as above, but work with certainsubrings, namely restricted products with respect to a formula ϕ ( x ) of the languageof rings (in a single variable x ), defined as the set of all f ∈ Q i ∈ I R i so that { i : R i | = ϕ ( f ( i )) } is cofinite. Provided that ϕ ( x ) defines a unital subring of each R i , the above subset is in fact a subring of Q i ∈ I R i (not in general definable).We obtain, for restricted products, results exactly analogous to those of [3] forproducts. Given ϕ ( x ) , we provide axioms in the language of rings augmented bya predicate F in ( x ) , and prove that any commutative unital ring satisfying these Mathematics Subject Classification.
Primary 03C10,03C60,11R56,11R42,11U05,11U09,Secondary 11S40,03C90. † Supported by a Leverhulme Emeritus Fellowship. axioms is elementarily equivalent to a restricted product, with respect to ϕ ( x ) , ofconnected rings. The standard interpretation of F in ( x ) in any Boolean algebra,and in particular in the Boolean algebra of idempotents of R is the ideal of finiteelements, i.e. finite unions of atoms.The canonical example of a such a restricted product, very important in numbertheory (see Cassels and Frohlich [1]), is A K , the ring of adeles over a number field K . Here I is the set of normalized absolute values v on K up to equivalence, R i isthe completion K v of K at v , and ϕ ( x ) is a formula of the language of rings thatdefines, uniformly for all v , the valuation ring O v of K v . (That there is such a ϕ ( x ) is nontrivial, and it is an important result that there is an ∃∀ -formula ϕ ( x ) that works uniformly for all O v , and hence for all adele rings uniformly in K , see[2] and [6]).In the case of adeles A K , the set of idempotents with finite support is definableby a formula of the language of rings independently of K (cf. [6], and a new proofgiven at the end of this paper in a ring-theoretic situation). Thus we can deriveaxioms in the language of rings for the adeles, uniformly in K .1. The Boolean algebra of idempotents of a ring
We shall denote the language of rings by L rings = { + , ., , } and the language ofBoolean algebras by L Boolean = {∨ , ∧ , ¬ , , } . We start by recalling the definitionof a restricted product of structures with respect to a formula (cf. [6], [7], [5]). Definition 1.1.
Let L be a language and ( M i ) i ∈ I a family of L -structures. Let ϕ ( x ) be a L rings -formula in the single variable x . The restricted direct productof M i with respect to ϕ (also called product restricted by ϕ ) is the subset of theproduct Q i ∈ I M i consisting of all f such that M i | = ϕ ( f ( i )) for all but finitelymany i ∈ I . We denote this restricted product by Q ( ϕ ) i ∈ I M i . It is a substructure of thegeneralized product defined by Feferman and Vaught in [9] provided that ϕ ( M i ) is a substructure of M i for all i ∈ I . The results in [9] and [6], [5], [7] yield generalquantifier eliminations for such restricted products, where L is any many-sortedlanguage. One can deduce, among other results, quantifier elimination for adeles,and results on definable subsets of adeles and their measures.1.1. Atoms and Stalks.
We follow as much as possible the development from Section 1 of [3].
Definition 1.2.
Let R be a commutative unital ring. The set { x : x = x } ofidempotents is a Boolean algebra, denoted by B , with operations e ∧ f = ef, XIOMS FOR RESTRICTED PRODUCTS 3 ¬ e = 1 − e, , ,e ∨ f = 1 − (1 − e )(1 − f ) = e + f − ef. It carries a partial ordering defined by e ≤ f ⇔ ef = e (which is L rings -definable).The atoms of B are the minimal idempotents (with respect to the ordering) thatare not equal to , . (In fact we assume = 1 ). Note that if R is a product, over an index set I , then B is isomorphic to theBoolean algebra of subsets of I via "characteristic functions". Lemma 1.1.
For any e in B we have R/ (1 − e ) R ∼ = eR ∼ = R e , where R e is thelocalization of R at { e n : n ≥ } .Proof. See Lemma 1 in [3]. (cid:3)
We call R e the stalk of R at e . Of special important are the R e for atoms e .Now we gradually impose axioms on R , in order to get a converse to Feferman-Vaught for restricted products. Axiom 1. B is atomic. Notes.
This holds if R is a restricted product of connected rings. One doesnot even need the restricting formula ϕ ( x ) to be definable. Moreover R andthe unrestricted product have the same idempotents. The basic example is A K embedded in Q v K v .Now we go through a series of consequences of the current axioms, and additionsof new axioms. Lemma 1.2. If f ∈ B , and f = 0 , then f = W { e : e an atom , e ≤ f } , (where W is union or supremum).Proof. This is Lemma 2 in [3]. (cid:3)
We turn to Boolean values and follow 1.3 of [3].
Definition 1.3.
Let Θ( x , . . . , x n ) be a formula of the language of rings, and f , . . . , f n ∈ R . Then [[Θ( f , . . . , f n )]] is defined as _ e { e : e an atom , R e | = Θ(( f ) e , . . . , ( f n ) e ) } provided W exists in B . Here ( f ) e is the natural image of f in R e (or, seen fromperspective of Lemma 1.1, f + (1 − e ) R ). J. DERAKHSHAN AND A. MACINTYRE
Axiom 2. [[Θ( f , . . . , f n )]] exists (as an element of B ). Notes. If R is a product of structures then B is complete, however completenessof a Boolean algebra is not a first-order property.Axiom 2 is a substitute for completeness (and follows from it).Axiom 2 is true in a restricted product of connected rings with respect to agiven formula ϕ (¯ x ) .1.2. Boolean Values and Patching.
The [[Φ( f , . . . , f n )]] are in B , and occur in [9] in the context of products, witha different notation. The [[ ... ]] notation comes from Boolean valued model theory[]. The next Lemmas come from 1.4 of [3]. Lemma 1.3.
Let Θ , Θ , Θ be L rings -formulas in the variables x , . . . , x n . Thenfor any f , . . . , f n ∈ R , • [[(Θ ∧ Θ )( f , . . . , f n )]] = [[Θ ( f , . . . , f n )]] ∧ [[Θ ( f , . . . , f n )]] , • [[( ¬ Θ)( f , . . . , f n )]] = ¬ [[Θ( f , . . . , f n )]] , • [[(Θ ∨ Θ )( f , . . . , f n )]] = [[Θ ( f , . . . , f n )]] ∨ [[Θ ( f , . . . , f n )]] .Proof. These statements are Lemmas 3-5 in [3] (cid:3)
These are some of the ingredients used in inductive proofs of result in [9].We add another axiom, taken from 1.4. of [3]
Axiom 3.
For any atomic formula Θ( x , . . . , x n ) of the language of rings, R | = Θ( f , . . . , f n ) ⇔ B | = [[Θ( f , . . . , f n )]] = 1 . This is evidently true in restricted products, no matter what ϕ ( x ) is.Now we fix a ϕ ( x ) , and aim for axioms true in restricted direct products R withrespect to ϕ ( x ) .We come now to a fundamental point. Classically the notion of restricted prod-uct appeals to the absolute notion of finite which is not, of course, first-order. Weare aiming for first-order axioms in some natural formalism. As already suggested,we are going to use an idea from Feferman -Vaught [9] of working with Booleanalgebras B with a distinguished subset F in , which in the case of the power setalgebra is the ideal of finite sets. In the case of a Boolean algebra of idempotents, F in will be the set of finite idempotents, as explained earlier. XIOMS FOR RESTRICTED PRODUCTS 5
We will shortly be concerned with other interpretations of a predicate symbolfor F in , indispensable for understanding nonstandard models of our axioms (andin particular nonstandard models of the theory of the adeles.But first we use provisional "axioms" where "finite" really means finite, and"finite idempotents " really mean finite idempotents, and "cofinite" really meanscofinite.We could avoid this ,and pass directly to the general case. But we prefer todiscuss a provisional axiom connected to the kind of patching used in [9] Axiom + . For all Θ( x , . . . , x n , w ) , f , . . . , f n , there is a g ∈ R such that if [[ ∃ w ( ϕ ( w ) ∧ Θ( f , . . . , f n , w ))]] is cofinite in [[ ∃ w Θ( f , . . . , f n , w )]] , then [[ ∃ w Θ( f , . . . , f n , w )]] is cofinite in [[Θ( f , . . . , f n , g )]] .This is clearly true in restricted products with respect to ϕ ( x ) (use Axiom ofChoice). Note. [3] has a simpler Axiom 4 for the unrestricted product case. That is notneeded here.
Note.
From now on, we will get involved with not only B , but with the ideal F in in B consisting of finite elements of B , i.e. finite unions of atoms.We have to enrich the first-order language of Boolean algebras by a 1-ary pred-icate symbol F in ( x ) . For our purposes B will be atomic as above, and F in ( x ) interpreted as the ideal of finite support idempotents. The interpretation of F in ( x ) in a Boolean algebra of sets, e.g. the powerset P ( I ) of a set I is the (Boolean)ideal of finite sets.However, note that Axiom + is not first-order. Any anxieties about this shouldbe removed by considering the result that the theory of the class of all infiniteatomic Boolean algebras in the language of Boolean algebras augmented by F in ( x ) is axiomatizable and complete (and admits quantifier elimination). This is provedfirst by Tarski but we give a new proof with explicit axioms in [8]. See also Section1.4 below. [8] contains a unified treatment that includes further expansions bypredicates for "congruence conditions on cardinality of finite sets".We return to this matter later, reformulating Axiom + in terms of F in ( x ) .1.3. Partitions.
J. DERAKHSHAN AND A. MACINTYRE
In order to sketch a proof of a useful generalization of [9] to our more restrictivesituation (rings R satisfying the axioms listed above) we need to review severalnotions of partition used in [9]. Notion 1.
In a Boolean algebra B a partition is a finite sequence < Y , . . . , Y m > of elements of B such that Y ∨ · · · ∨ Y m = 1 and Y i ∧ Y j = 0 if i = j . (We do not insist that each Y i = 0 , but do insist that thesequence is finite).We note that in the definition of partition "finite" will always mean finite. Notion 2.
For a first-order language L , a fixed m , and L -formulas Θ ( x , . . . , x m ) , . . . , Θ m ( x , . . . , x m ) the sequence < Θ , . . . , Θ m > is a partition if the formulas Θ ∨ · · · ∨ Θ m and ¬ (Θ i ∧ Θ j ) (where i = j ) are logically valid.(This is of course ultimately a special case of Notion 1).The basic lemmas about disjunctive normal form in propositional calculus, whenapplied to formulas ψ ( x , . . . , x m ) , . . . , ψ l ( x , . . . , x m ) give constructively a partition whose elements are propositional combinations ofthe ψ i ’s. This is used crucially in [9].The final result we need before sketching [9] for all our rings is. Lemma 1.4. (Analogue of Lemma 6 in [3] ) Suppose Y , . . . , Y k is a partition of B . Suppose the sequence < Θ ( x , . . . , x m , x m +1 ) , . . . , Θ k ( x , . . . , x m , x m +1 ) > is a partition. Suppose f , . . . , f m ∈ R and Y j ≤ [[ ∃ x m +1 Θ j ( f , . . . , f m , x m +1 )]] for each j . Suppose in addition that for each jY j ∧ ¬ [[ ∃ x m +1 ϕ ( x m +1 ) ∧ Θ j ( f , . . . , f m , x m +1 )]] is finite. Then there is a g in R so that Y j ⊆ [[Θ j ( f , . . . , f m , g )]] for all j .Proof. Apply Axiom + to each Y j and Θ j , f , . . . , f m to get g j ∈ R so that Y j ⊆ [[Θ j ( f , . . . , f m , g j )]] . Now let g = P j g j .e j , where e j is the idempotentcorresponding to Y j . (cid:3) XIOMS FOR RESTRICTED PRODUCTS 7
Note.
Later we re-do this in terms of
F in (subject to Tarski’s axioms).1.4.
The Augmented Boolean Formalism.
We adjoin to the first-order language for Boolean algebras L Boolean a unary pred-icate symbol
F in ( x ) . Denote the augmented language by L finBoolean . The standardinterpretation of F in ( x ) in a Boolean algebra is the ideal F in of finite elements,i.e. finite unions of atoms. However, the class of such ( B , F in ) is not elementary,and it is important for us to give a computable set of axioms complete up to spec-ifying the the number of atoms below an element. We do this in [8] (as part ofa new expansion of L Boolean ), but the original work was done by Tarski (see [9]).Here are the essential points.Let T be the theory of infinite atomic Boolean algebras in the language L Boolean expanded by the definable relations C k ( x ) ( k = 1 , , . . . ) with the interpretationthat x has at least k atoms α ≤ x . Tarski proved that in this language the theoryof infinite atomic Boolean algebras is complete, admits quantifier elimination, andis decidable (see [9], [8]). The axioms for this theory state that the models areinfinite Boolean algebras and every nonzero element has an atom below it. Let ♯ ( x ) denote the number of atoms α such that α ≤ x .Now we further expand the given language by adding the predicate F in ( x ) withthe above interpretation in any Boolean algebra, and obtain L finBoolean . We add tothe axioms of T the axioms stating that F in is a proper ideal, the sentence ∀ x ( ¬ F in ( x ) ⇒ ( ∃ y )( y < x ∧ ¬ F in ( y ) ∧ ¬ F in ( x − y ))) . and, for each n < ω , the sentence ∀ x ( ♯ ( x ) ≤ n ⇒ F in ( x )) . This defines an L finBoolean -theory T fin . Theorem 1.1. [9] , [8] The theory T fin of infinite atomic Boolean algebras with theset of finite sets distinguished is complete, decidable and has quantifier eliminationwith respect to all the C n , ( n ≥ , and F in (i.e. in the language L finBoolean ).The axioms required for completeness are the axioms of T together with sentencesexpressing that F in is a proper ideal, the sentence ∀ x ( ¬ F in ( x ) ⇒ ( ∃ y )( y < x ∧ ¬ F in ( y ) ∧ ¬ F in ( x − y ))) . and, for each n < ω , the sentence ∀ x ( ♯ ( x ) ≤ n ⇒ F in ( x )) . Note:
This is important for measurability of definable sets in adele rings as in[6].
Note:
In [8] we prove that F in is not definable in the language of the theory T . J. DERAKHSHAN AND A. MACINTYRE
Modifying the Preceding (provisional) Axioms for R . Our axioms given so far are not first-order. To rectify this, we first do thefollowing. Work with rings R together with a distinguished ideal in B . In theadelic cases this ideal will be the ideal of finite idempotents, but we will also beinterested in other ideals. We write F in for the distinguished ideal. We modifyaxiom + to Axiom F in (which is still not first-order). We let F in be the set ofrealizations in R of the predicate F in ( x ) . Axiom fin . There is an ideal F in in B so that ( B , F in ) | = T fin , and such thatfor all Θ( x , . . . , x n , w ) , f , . . . , f n there is a g ∈ R such that if [[ ∃ w Θ( f , . . . , f n , w )]] ∩ ¬ [[ ∃ w ( ϕ ( w ) ∧ Θ( f , . . . , f n , w ))]] ∈ F in, then [[ ∃ w Θ( f , . . . , f n , w )]] ∩ ¬ [[Θ( f , . . . , f n , g )]] ∈ F in. This is clearly true in classical products restricted by ϕ ( x ) (use Axiom ofChoice).In Lemma 1.4 we need to change "finite" to "in F in ", and the proof goesthrough, getting
Lemma 1.5.
Suppose Y , . . . , Y m is a partition of B . Suppose the sequence < Θ ( x , . . . , x k , x k +1 ) , . . . , Θ m ( x , . . . , x k , x k +1 ) > is a partition. Suppose f , . . . , f k ∈ R and Y j ⊆ [[ ∃ x k +1 Θ j ( f , . . . , f k , x k +1 )]] for each j . Suppose in addition that for each jY j \ [[ ∃ x k +1 ϕ ( x k +1 ) ∧ Θ j ( f , . . . , f k , x k +1 )]] ∈ F in Then there is a g in R so that Y j ⊆ [[Θ j ( f , . . . , f k , g )]] for all j . We now have Axioms 1-3 and Axiom fin . Note that ϕ ( x ) , the restrictingformula, is fixed.There is one last Axiom 5. Axiom 5. ∀ x ( F in ([[ ¬ ϕ ( x )]])) .Call the resulting axiom set A ϕ , axioms for ϕ -restricted products.We have given axioms for pairs ( R, F in ) , and we shall next prove that we havea Feferman-Vaught type theorem. XIOMS FOR RESTRICTED PRODUCTS 9 The Feferman-Vaught Theorem
The Main Theorem.Theorem 2.1.
Let ϕ (¯ x ) be an L -formula. Let R a commutative unital ring satis-fying the axioms A ϕ . Then for each L rings -formula Θ( x , . . . , x m ) there is, by aneffective procedure, a partition < Θ ( x , . . . , x m ) , . . . , Θ k ( x , . . . , x m ) > of L rings -formulas, and an L finBoolean -formula ψ ( y , . . . , y k ) such that for all f , . . . , f m in R R | = Θ( f , . . . , f m ) ⇔ ( B , F in ) | = ψ ([[Θ ( f , . . . , f m ) , . . . , Θ k ( f , . . . , f m )]]) . Proof.
In [9], there is a standard inductive proof for this by induction on thecomplexity of Θ for the case of generalized products. These are the products thatare equipped with extra relations making it into a generalized product. That proofcan be modified to go through for the case of restricted products with respectto a given formula ϕ (which is a substructure of a generalized product). Thismodification can be made to work in the case of our rings R .If Θ( x , . . . , x m ) is a quantifier-free formula, then we can take the Boolean for-mula [[Θ( x , . . . , x m )]] = 1 . Then for all f , . . . , f m ∈ R , R | = Θ( f , . . . , f m ) ⇔ B | = [[Θ( f , . . . , f m ) = 1]] . Now suppose that Θ is of the form ∃ x m +1 Θ ∗ ( x , . . . , x m , x m +1 ) , assuming the result known for Θ ∗ .Now for any f , . . . , f m ∈ R , R | = ∃ x m +1 Θ ∗ ( f , . . . , f m , x m +1 ) if and only if ( ∗ ) for some g ∈ R R | = Θ ∗ ( f , . . . , f m , g ) . By the inductive hypothesis, there is a partition < θ ′ , . . . , θ ′ k ′ > and a Booleanformula Φ ′ (both associated to Θ ∗ ) such that, ( ∗ ) is equivalent to ( ∗ ) for some g ∈ RR | = Φ ′ ([[ θ ′ ( f , . . . , f m ) , g ]] , . . . , [[ θ ′ k ′ ( f , . . . , f m ) , g ]]) . Now we us put k = k ′ , and θ j = ∃ x k +1 θ ′ j , ≤ j ≤ k ′ , and define the following Boolean formula Φ( z , . . . , z k ) = ∃ y , . . . ∃ y k P art k ( y , . . . , y k ) ^ j y j ≤ z j ⇔ F in ( y j \ [[ ∃ x k +1 ϕ ( x k +1 ) ∧ θ ′ j ( f , . . . , f k , x k +1 )]]) ∧ Φ ′ ( y , . . . , y k ) . We show that ( ∗ ) is equivalent to ( ∗ ) B | = Φ([[ θ ( f , . . . , f k )]] , . . . , [[ θ k ( f , . . . , f k )]]) . Assume ( ∗ ) . Define y j = [[ θ ′ j ( f , . . . , f k , g )]] . Then for each jy j ≤ [[ θ j ( f , . . . , f k )]] . Since < θ , . . . , θ k > is a partition, P art k ( y , . . . , y k ) holds and ( ∗ ) follows.Conversely, suppose that ( ∗ ) holds. Then there are elements b j that form apartition of B , and for each j we have b j ≤ [[ ∃ x k +1 Θ j ( f , . . . , f k , x k +1 )]] , and such that B | = F in ( b j \ [[ ∃ x k +1 ϕ ( x k +1 ) ∧ Θ j ( f , . . . , f k , x k +1 )]]) and B | = Φ ′ ( b , . . . , b k ) . By Lemma 1.5 there is g ∈ R such that for all jb j ⊆ [[ θ ′ j ( f , . . . , f k , g )]] . Since < b , . . . , b k > and < θ ′ , . . . , θ ′ k > are both partitions, for all j we have b j = [[ θ ′ j ( f , . . . , f k , g )]] . This proves ( ∗ ) . (cid:3) Corollary 2.1.
For R as above, with restricting formula ϕ , R ≡ ( ϕ ) Y e atom of B R e , the restricted product with respect to ϕ .Proof. Both rings have the same idempotents, the same ideal F in , and same R e ( e an atom), the same ϕ , and satisfy the axioms A ϕ . (cid:3) Note the effectivity and uniformity of Theorem 2.1 in ϕ and all rings satisfyingthe axioms A ϕ . XIOMS FOR RESTRICTED PRODUCTS 11
Ring-Theoretic Definability of F in . In [6] we show that the ideal F in is L rings -definable uniformly in all A K , K anumber field. In fact there is ring-theoretical definition of F in for a large class ofrings satisfying our axioms (and the definition is in a clear sense uniform in ϕ ).We have to require the following of R and ϕ . ( ♯ ): Suppose e ∈ F in . Then then there are g, h ∈ R so that [[ e = 1]] ⊆ [[ gh = 1 ∧ ϕ ( g ) ∧ ¬ ϕ ( h )]] . Note that this is true when R = A Q .Now we proceed to an L rings -definition of F in assuming ♯ .Suppose first f ∈ R is an idempotent and [[ f = 1]] / ∈ F in . Then there is no g, h ∈ R with [[ f = 1]] ⊆ [[ gh = 1 ∧ ϕ ( g ) ∧ ¬ ϕ ( h )]] ⊆ [[ ¬ ϕ ( h )]] ∈ F in.
On the other hand, suppose [[ f = 1]] ∈ F in . Then by ( ♯ ) there exist g, h ∈ R with [[ f = 1]] ⊆ [[ gh = 1 ∧ ϕ ( g ) ∧ ¬ ϕ ( h )]] . So we have.
Theorem 2.2.
Suppose R satisfies ♯ . Then F in is definable.Proof. e = [[ e = 1]] , so e ∈ F in ⇔ ∃ g ∃ h ( e ≤ [[ gh = 1 ∧ ϕ ( g ) ∧ ¬ ϕ ( h )]]) . (cid:3) References [1]
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St Hilda’s College, University of Oxford, Cowley Place, Oxford OX4 1DY,UK
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