Ultraproducts for Algebraists

A.3

Ultraproducts for Algebraists PAUL C . EKLOF

Contents Introduction

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BASICS Filters . . . . . . . Reduced products . . . The fundamental theorem. Ultraproducts as functors .

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106

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106 108 111 114

COMPACTNESS 5. An ultraproduct version of the Compactness Theorem. . . . . . . . 6. Embedding theorems . . . . . . . . . . . . . . . . . . . 7. Bounds in polynomial ideals. . . . . . . . . . . . . . . . .

118 120 124

SATURATION 8. Ultraproducts which are w,-saturated. . . . . . . . . . . . . . 9. Ultraproducts of valued fields . . . . . . . . . . . . . . . . 10. Saturated models. . . . . . . . . . . . . . . . . . . . .

127 132 134

1. 2. 3. 4.

References

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HANDBOOK OF MATHEMATICAL LOGIC Edited by J. Banvise

0 North-Holland Publishing Company, 1977

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Introduction The ultraproduct construction is an algebraic operation whose importance derives from its model-theoretic properties. The algebraic character of the construction makes it a very attractive (though not essential) tool to employ in giving an account of applications of model theory to algebra. This paper is a survey of the basic properties of ultraproducts and of some of their applications to algebra. The prerequisites for reading the paper are familiarity with the fundamental definitions of model-theory given in the previous chapters, and with the definitions of “category”, “functor”, and “natural transformation”. Moreover, since the applications to algebra are taken from such diverse areas as group theory, ring theory, algebraic geometry, universal algebra and algebraic number theory, we have found it necessary to assume an acquaintance with the algebraic notions and results used in the examples. (Usually we have given references for definitions and results not found in LANG[1971].) However, the reader may skip over a particular algebraic example without any loss of understanding of later material (except possibly of related algebraic examples). The paper is divided into three parts. The first part, Basics, gives the definitions of ultrafilters and ultraproducts (as well as of a generalization of the ultraproduct called the reduced product) and proves the “fundamental theorem of ultraproducts”. In the last section of the first part we discuss the functorial properties of ultraproducts. The rest of the results are given in two parts, Compactness and Saturation, whose titles refer to the key model-theoretic properties of ultraproducts used in proving these results. Some of the deepest applications of ultraproducts to algebra are only mentioned here, but references are given. Also, because the emphasis in this paper is on algebraic applications of ultraproducts, we have not discussed at all many interesting results about ultraproducts which do not fall under this heading. For more about ultraproducts we refer the reader to the excellent survey articles by KEISLER[1965] and CHANG [I9671 and to the textbooks CHANG-KEISLER [I9731 and BELLand SLOMSON [1969].

1. Filters

If Z is a non-empty set, a filter over I is a set D of subsets of Z such that: (i) 0 $ Z D , I E D ;

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FILTERS

(ii) if X , Y E D, then X f l Y E D ; (iii) if X E D and X C Y C I , then Y E D. For example, if Y is a non-empty subset of I, the set { X C I I Y C X } is a filter over I, called the principalfilter generated by Y ; we denote it by ( Y ) . If I is finite, every filter over I is principal (generated by {X 1 X E D}). If I is infinite, an example of a non-principal filter over I is the cofinitefilter C = { X c I1 I - X is finite}. Notice that by (i) and (ii), a filter D over I has the finite intersection property (FIP), i.e., the intersection of any finite set of elements of D is non-empty. Obviously any subset of D also has FIP. Conversely, if S i s a set of subsets of I which has the FIP, then S is a subset of a filter over I ; in fact

n

D

={YC I

1 X I f l . . . fI X . c Y for some X I , .. ., X , E S }

is a filter containing S. A filter D over I is called an ultrafilter over I if for every X C I , either X E D or ( I - X ) E D . 1.1. PROPOSITION. A filter over I is an ultrafilter i f and only if it is a maximal filter over I . Proof. Suppose D is an ultrafilter over I and suppose E is a filter over I such that D C E and D # E. Then X E E - D for some X C I. Since X E D we have I - X E D by definition of an ultrafilter. But then I - X E E, which is impossible since { X , I - X } C E does not have the FIP. Conversely, suppose D is a maximal filter and let X C I such that X g D. Then D U {I- X } has FIP so it is a subset of a filter E. By the maximality of D, E must equal D and so I - X E D. 0

An application of Zorn’s Lemma yields the following. 1.2. COROLLARY. If S is a set of subsets of I which has the finite intersection property, then S is contained in a n ultrafilter over I. 1.3. PROPOSITION. If D is an ultrafilter over I, X E D , and X = Y IU U Y,, then for some i, Yi E D.

---

Proof. If not, then by definition of ultrafilter, I - Y, is a member of D for all i = 1,. . ., n. But then by (ii) of the definition of a filter, O = X fl ( I - Y 1 f) l . fl ( I - Y.) is a member of D, which contradicts (i). 0

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A principal ultrafilter is an ultrafilter which is a principal filter. If D is a filter over I and { x } E D for some x E I, then D is a principal ultrafilter over I generated by {x}; in fact, Y E D iff Y f l { x } E D iff I x } C Y. Conversely, if D is a principal ultrafilter, then D must be generated by a singleton; for, otherwise, if D is generated by Y and there exists O f Y ' s Y, then the filter generated by Y' is a proper extension of D, contradicting 1.1. 0

1.4. PROPOSITION. For any ultrafilter D over I, D is non-principal if and only if D contains C, the cofinite filter. PROOF.If D is a non-principal ultrafilter over I and X = { x l , . . ., x,} is a finite subset of I, then by the above remarks { x i }fi? D for all i = 1,. . ., n. Hence I - { x , } E D for all i = 1, .. ., n so , ? = , I - { x i } = ( I - X ) E D. Therefore C D. Conversely if C C D, then {x} fi? D for all x E I (since I - { x } E C), so by the remarks preceding the theorem, D is nonprincipal. 0

n

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1.5. COROLLARY. Let S be a set of subsets of I such that XI f l * * n X , is infinite for all X , , . . . , X n E S. Then S is contained in a non-principal ultrafilter over I . PROOF.It follows from the hypothesis on S that C U S has the FIP, so we can apply 1.2 and 1.4. 0

2. Reduced products

Let L be a first-order language; that is, L is a collection of relation, function, and constant symbols. We refer to 3.1 of Chapter A . l for the definition of a model (or structure) 91 for L. (When we talk about specific algebraic systems we will sometimes follow algebraic custom and denote t h e model 8 by its universe A.) If 2 is a consistent set of sentences of L let A ( 2 )denote the class of all models of 2 i.e. the class of all models '$l for L such that every sentence of 2 is true in 8 , denoted 8 I= 2.By an abuse of notation we shall write A(L) instead of A(0);thus A(L) is the class of all models for L. If 91, 2 '3 E A(L), a homomorphism 77 from ?l to 2'3 is a set function 7 : A + B which preserves all the relations, functions, and constants of L. For example, if L = { , 5 ,0} (where is a binary function symbol, 5 is a

+

+

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REDUCED PRODUCIS

binary relation symbol and 0 is a constant symbol) then v : a+ 8 is a homomorphism if and only if ~ ( 0= ~~ ( 0) m ) and for all a l , u2E A v(a,+a2)= v(aJ+v(a2);

a15

a2

*

v(a1)S 77(a2).

As usual, an isomorphism is a homomorphism which has an inverse homomorphism, and an embedding is a homomorphism which is an isomorphism onto its range. If (?I,: i E I) is a family of elements of A(L) indexed by a set Z, denote by II,2l, the direct product of the family; that is nI?l,is the model for L whose universe is the direct product IIIA, of the sets A, and whose relations, functions and constants are defined “componentwise”. For example, if ?I, = (A,, + , 5 , , O , ) , then IIl?l, = (n,A,, i, 2 , 6 ) where 6 ( i )= 0, for all i E I, and for any f, g E IIIA,,(fi g ) ( i ) = f ( i ) + , g ( i ) ; and g iff f ( i ) 5 , g ( i ) for all i E I. If (a,: i E Z) is a family indexed by Z and X is a subset of Z we shall understand by IIx21, the direct product of the family (a, : i E X). Notice that if Y C X C Z , the canonical projection 7rxv :IIxA,-*IIuA, is a homomorphism. Let D be a filter over a set I. Observe that if we define a partial ordering on D by X IY e X 2 Y, then (D, I) is a directed set (i.e. for any X , Y E D there exists 2 E D such that X 5 Z and Y 5 2 ) .If (el, : i E I) is a family of models for L indexed by I, the reduced product of (a, : i E I ) modulo D, denoted nD?l,, is defined to be the direct limit (in A(L)) of the directed system { r x v : n,?l, -* II,?l, I X , Y E D, X 5 Y}. If D is an ultrafilter, IID?l, is called the ultraproduct of (?I,: i E I ) modulo D. If 8, = ?! for all i E I, we write nD?! instead of OD?!, and call it the ultrapower of 91 modulo D. Although the shortest approach to the definition of reduced products is via the notion of direct limit, this approach is perhaps misleading since it is the concrete construction of the direct limit rather than its universal mapping properties which will be of importance in the sequel. So let us describe the structure of IID21,more explicitly. The following description of the reduced product follows easily from the usual construction of direct limit, given the observation that for every X E D the projection rlxis surjective, so that every element of the direct limit is represented by an element of II,A,. Those readers who are not comfortable with the notion of direct limits may take the following as the definition of reduced products. Define an equivalence relation = on n l A , as follows: If f, g E HlA,,

fz

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then f = g if and only if rrx (f) = r r x ( g )for some X E D i.e. f = g if and only if { i E 11 f ( i ) = g ( i ) } is an element of D. The universe of nD%,is the set nDA, of equivalence classes of elements of IIIA,. Denote the equivalence class of f E nrA, by f D . The relations on n,?I, are defined by the condition that a relation holds between elements of nD%, if and only if it holds on a set of components which belongs to D ; similarly for functions. For example, if ?= I, (A,,+,,5 ,,o,) and f D , gD. hD E nDA,, then f D 5 go in n D Y , if and Only if { i E 11 f ( i ) s , g ( i ) } ED. (Equivalently, f D 5 gD if and only if there exists X E D such that r r x ( fI ) r I x ( g )in n,'?!,.) Similarly, f D + gD = hD in nD?f, if and only if { i E I I f ( i ) + g(i) = h ( i ) }E D. And the interpretation of 0 in n,'?l, is 6, where { i E I 1 6 ( i ) = O , } E D. It is easy to verify, using t h e properties of filters, that 5 , +, and 0 on nD'?l,are well-defined. The following lemma justifies the similarity of our notation for direct products and reduced products. It is an immediate consequence of the fact that the directed set ((X), I) has X as a largest element. 2.1. LEMMA.If ( X ) is the principal filter over I generated by X , then the reduced product of (%, : i E I ) modulo ( X ) is isomorphic to the directproduct of (8, : i E X ) (i.e. ncx,A, = n,A,). It is ultraproducts rather than reduced products in general that are of most interest to logicians since ultraproducts preserve logical properties of the family (?f , : i E I ) (in a sense which is made precise in the main theorem of the next section). But before proceeding to that, let us mention that in the case when the models are division rings, an alternate construction of the reduced product may be given. Let ( R , : i E I ) be a family of division rings. For any f E n,R,, let Z ( f )= { i E I1 f ( i ) = 0). Notice that for any f , g E n,R,, we have Z ( f ) Z ( g ) if and only if there exists h , i n r R , such that h,f = g if and only if there exists h , E n,R, such that fhz = g. (Here we use the fact that the R,'s are division rings.) It follows that every (left or right) ideal of n l R , is two-sided. If N is a subset of n r R , , let Z ( N ) = { Z ( f ) :f E N } . 2.2. THEOREM (KOCHEN[1961]). Let ( R , : i E I ) be a family of division rings. If N is an ideal of n ,R,, then Z ( N ) is a filter over I. The mapping, 2, which assigns N to Z ( N ) is a one-one inclusion-preserving correspondence

between proper ideals of N and filters over I. Under this mapping, principal ideals correspond to principal filters and maximal ideals correspond to

'

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ultrafilters. Moreover, for any ideal N of n r R , the quotient ring n I R i / Nis isomorphic to the reduced product n Z ( N , R i . PROOF.If S is a subset of I, let SE n,R, denote the “characteristic function” of S i.e. S : I --+ (0,l) C R such that s ( x ) = 0 if and only if x E S. Notice that if N is an ideal of n r R i and f E n,R,, then f E N if and only if Z ( f ) E N . It follows that for any f, g E N , Z ( f ) f l Z ( g )E Z ( N ) , since

(A simpler expression is possible if none of the Ri have characteristic 2 . ) Thus Z ( N ) satisfies (ii) of the definition of a filter; parts (i) and (iii) of the definition follow easily from the definition of Z ( N ) and the remarks preceding the theorem. Hence Z ( N ) is a filter. If D is a filter over Z let ND = { f E n l R i 1 Zcf)E D } ; then N is an ideal since Z ( f - g) 2 Z ( f ) n Z ( g ) and Z c f g ) 2 Z c f ) ; N is proper since 0 E D. The mapping which takes D to ND is inverse to the mapping Z and hence Z is a one-one correspondence between ideals of rkrR, and filters over I. Since Z is clearly inclusion preserving, maximal ideals correspond to maximal filters i.e. ultrafilters (see Proposition 1.1). It is also easy to verify that the ideal N is generated by f if and only if Z ( N ) is generated by Z ( f ) . Finally, the function which takes the coset f + N in n r R i / Nto f D E n Z ( N ) R i is an isomorphism of rings. 0 2.3. COROLLARY. Z f ( R i: i E Z) is a family of division rings and N is a prime ideal of n,Ri, then N is a maximal ideal and n I R i / Nis a d h o n ring.

PROOF.To prove that N is maximal, it suffices to prove Z ( N ) is an ultrafilter. Let X-C- 1. If X$Z Z ( N ) and Z - X$Z Z ( N ) ,then and F X a r e not in N . But X ( l - X)= 0 E N, a contradiction. Therefore Z ( N ) is an ultrafilter. To see that n r R i / N is a division ring, let f E n r R i such that f $ Z N . Then Z ( f ) !Zi Z ( N ) so I - Z ( f ) E Z ( N ) . I f ’ g E n l R i such that g ( i ) = f ( i ) - ’ for all i E Z - Z ( N ) , then { i E Z I g ( i ) f ( i ) = 1) = Z - Zcf)E N . Therefore by the remarks preceding Theorem 2.2, 1 - gf E N, i.e. f is a unit in n r R i / N . 0

x

3. The fundamental theorem As a result of Corollary 2.3 we know that an ultraproduct of division rings is a division ring (while a reduced product of division rings which is

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not an ultraproduct will not be a division ring). This is a special case of the fundamental theorem of ultraproducts which says that elementary properties of the family (?(,: i € I ) are preserved under ultraproducts. More precisely, the fundamental theorem, or Lo<' Theorem, is the following.

3.1. THEOREM (kos [1955]). Let (?Ii:i E I ) be a family of models for L and let D be an ultrafilter over I. Then for any formula cp (x,, . . ., x.) of L and any elements g I , . . ., g " of fl ,?1,,

nD?l,i=cp[gb*-*g;;] if and only if { i € I ~ ? 1 , C c p [ g ' ( i ) . . . g n ( i ) ] }DE.

Before proving the fundamental theorem, let us consider some consequences of i t . For the case when cp has no free variables we obtain t h e following. 3.2. COROLLARY. For any ultraproduct if and only i f { i E I / 91, I= cp} E D.

n

?I, and any sentence cp, II, ?I I=cp

3.3. COROLLARY. If 91, is in A ( Z ) for each i E I, then the ultraproduct n,?l, is in A ( 2 ) .

Z

A subclass of A ( L ) which is of the form A ( Z ) for some set of sentences

of L is called (first-order) axiomatizable, or elementary. Corollary 3.3 says that any axiomatizable class is closed under ultraproducts. For example, an ultraproduct of groups (semigroups; rings; commutative rings; fields; algebraically closed fields; division rings; formally real fields; real closed fields; Lie algebras; Boolean algebras; etc.) is a group (semigroup; ring; commutative ring; field; algebraically closed field; division ring; formally real field; real closed field; Lie algebra; Boolean algebra; etc.) because the class in question is axiomatizable. Proving that a class is not closed under ultraproducts is a useful method of proving that a class is not axiomatizable. We give an example. A different proof is given in 2.3 of Chapter A . l . It is instructive to compare the two proofs. 3.4. COROLLARY. Let L = { + ,0} be the language of abelian groups (i.e. L has a binary function symbol, + , and a constant symbol.0). The class of torsion abelian groups is nor first-order axiomatizable.

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PROOF.If suffices to produce an ultraproduct of torsion abelian groups which is not torsion. Let I = w and let A, be the cyclic group of order n + 1 for each n E I. We claim that if D is a non-principal ultrafilter over I, then IIDA, is not torsion. Let f E n,A, be such that f(n) has order n + 1 for each n E I. For any m > 0, if cpm (x) is the formula (x

+x +

* * *

+ x = 0)

(where x occurs m times), then by 3.1, II,A.Fqm[fD] since

is a finite set and hence is not in D by 1.4. Therefore fD is not of finite order. (Instead of appealing to 3.1 we could verify directly from the definitions that m f D # o in nDA. for any m # O . ) Hence nDA, is not torsion. 0 This method of proving that a class is not axiomatizable will not always work since-as we shall see in Section 4-there are classes closed under ultraproducts which are not axiomatizable. See Section 5 for an ultraproduct method of proving that a class is not finitely axiomatizable. Now let us prove 3.1.

PROOFOF THEOREM 3.1. For the sake of concreteness we shall prove the theorem for our standard example, ?I, = (A,, + ,, 5 ,,O,>, but the argument is perfectly general. The proof is by induction on the formula cp. Consider first the simplest kinds of atomic formulas: x , Ix2; x I + x 2 = x,; or x I = 0. If cp is one of these, then the result follows from the definition of the ultraproduct. The most general types of atomic formulas are t l It2 and t , = t2 where t l and t2 are terms built up from the function symbol + and the constant symbol 0 [e.g. tl = ((xl + x2) + ((0 + x,) + xl))+ ((x4 + x2) + O)]. In this case, the desired result follows from the first case above once we prove, by induction on the construction of a term t ( x , -x,), that trlD:ll,[gb. * .g>] = h, if andonly if { i E I / t Y , , [ g l ( i ) ...g " ( i ) ] = h(i)}E D. We leave thedetails to the reader: the initial cases (t = x,, + x , ~t; = x, ; or t = 0) follow from the definition of t h e ultraproduct. (See 3.7 in Chapter A . l for the definition of h, there written t".) Now suppose cp = i $(xl * x,) and suppose the theorem proved for $. Then

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(The next to last equivalence uses the fact that D is an ultrafilter, not simply a filter.) If cp = A $z, and the theorem is assumed for $, and $*,a similar argument proves the result for cp; here, the key fact is that for any filter D, X n Y E D if and only if X E D and Y E D. Finally, if cp = 3xo$(xl. * * x n ) , then nD%, I= cp [g b . g b ] e there exists there exists g o € H , % , g " E nf3 , such that HD%,!= $[goD, gb,. . . , g:] such that

+

{ i E Il?l,I=$lg"(i),g';..,g"(i)])E D { i E I(?[,I=cp[g'(i ) . . . g " ( i ) ] } E D.

+

Conversely, if { i E I1 ?I, I= cp [ g ' ( i ) . * . g " ( i ) ] }= X E D, then there exists g"E D,A, such that for every i E X , ?I, I= $ [ g " ( i ) , g ' ( i ) l , . . ., g " ( i ) ] ; hence { i E I 1 91, I= $ [ g " ( i ) , g ' ( i ) , . . ., g " ( i ) ] } E D. 0 Because of the importance of the fundamental theorem, from now on we shall confine o u r attention to ultraproducts (and ultrapowers) rather than arbitrary reduced products. 4. Ultraproducts as functors Thoughout the paper we shall assume that all the classes of models we consider are closed under isomorphism. If d is a subclass of .U(L), we shall also denote by d the cutegory whose objects are the elements of d and whose morphisms are all t h e homomorphisms between elements of d.We shall also consider the category d',whose objects are the indexed families (a,: i E Z) of objects of d and whose morphisms from (?I,: i E I) to (23,: i E I) are the indexed families ( 7 , : i E I) of homomorphisms 7 , : 91, + 23,. Since these are the only categories we consider, we shall freely interchange the words "class" and "category" in referrring to d or d'. If ( 7 , : i E I) is a family of homomorphisms 7 , : ?lt +8,,let nDv,: H D % , + n D B , be the function defined by ( n D q ) ( f D ) = gD, where g ( i ) = T , ( f ( i ) ) . It is easy to check that nD7, is a well-defined homomorphism called the ultraproduct of the 7 , modulo D. If 7 , ~=~ 7 : '3 +2'3 . for all i E I, then U D 7 , is denoted by and called

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the ultrapowerof 77 modulo D ; it is a homomorpnism from nD?l to n,a. It is now a routine exercise in the definitions to verify the following result. The notion of elementary embedding is defined in Chapter A.2. 4.1. THEOREM. If D is an ultrafilter over I and d is a subclass of A(L)

closed under ultrapowers (resp. ultraproducts), then the ultrapower (resp. ultraproduct) n, is a functor from d to d (resp. from d ’to d ) . 4.2. THEOREM. The ultrapower functor on d preserves embeddings and

elementary embeddings. Similarly the ultraproduct functor takes a familx ( 7 ,: i E I ) of embeddings (resp. elementary embeddings) to an embedding

(resp. elementary embedding). PROOF.If suffices to prove t h e result for ultraproducts. Now by definition 77, : 91, + 8 , is an embedding iff q, : 41, + ran q, is an isomorphism. Thus if ( 7 7 , : i E I ) is a family of embeddings, nDq, is an embedding because functors preserve isomorphisms and the range of n,q, = n, ran 7,.For the case of elementary embeddings we appeal to the fundamental theorem (3.1).Supposecp(xl...x.)isaformulaofLandgb,..., g;En,?l,.Then

One useful aspect of t h e ultraproduct construction is that-for a fixed ultrafilter-it provides a uniform way of defining functors on different categories. In order to give a formal expression of this uniformity we need a notion of “forgetful functor”. Let L, L’ be first-order languages such that L CL’. Recall that a model for L’ is a pair 3’ = (A’,$’) where 4‘assignsan interpretation of each symbol of L’. If ?l’ is a model for L’, let RLtL(?l’) be the model 3 = (A’,$) for L where 4 is the restriction of $‘to the symbols of L. R L * L ( ? I ’ ) is called the reduct of 91’ to L. If 77 :?1’+B’ is a homomorphism of models for L’, R L . . L ( q ) is the same set function 77 regarded as a homomorphism of R , . . , - ( ? 1 ’ ) to R L . L ( B ’ ) . It is easily seen that R L ’ L is a functor from A(L’) to A(L) called the forgetful functor. (We shall drop the subscripts on R where, in context, there is no ambiguity.) For example, if L is the empty set of symbols and L C L‘, then A(L) is the category of sets and R L . L is t h e familiar underlying set functor from A(L‘) to A(L).

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If R L Ltakes the subclass d'of A(L) to the subclass d of A ( L ) we shall also denote by RL.Lthe restriction of R c L to d and call it the forgetful functor from d'to d.For example there is a forgetful functor R L f L from the class of ordered abelian groups to the class of abelian groups. (Here L = { + , O } and L ' = { + , <,O}.) The next result is now an easy consequence of the definitions. 4.3. THEOREM. Let D be an ultrafilter over I and let L and L' be first-order

languages such that L C L'. Then the following diagrams (for the ultrapower and ultraproduct respectively) are commutative.

A(L')-A(L)

R LL

A&')

Ju(L)

Theorem 4.3 is useful in proving that certain classes are closed under ultraproducts, even when we do not know if the class is axiomatizable. Let us say that a class d c A ( L ) is pseudo-elementary if there is a forgetful functor R = R,., such that .sd is t h e image under R of an elementary (i.e. axiomatizable) class d'C A(L'). 4.4. COROLLARY. A pseudo -elementary class is closed under ultraproducts.

PROOF.Maintaining the notation of the definition, let us prove that for any family (?I,: i E I) in d,n,?f, is in d.By hypothesis, each '?I, is of the form R(?I:) for some 91: in d'.But then by Corollary 3.3 and Theorem 4.3 we have

nD?x= ~ I ~ R ( Y I :R(n,?r:)E )=

R(&')c~.

4.5. THEOREM. Let n E w. I f d is the class of all groups which are isomorphic to GL(n, F ) for some field F, then d is closed under ultra-

products. PROOF.Here we use G L ( n , F) to denote the group, under multiplication, of the invertible n x n matrices over F. By 4.4 it suffices to prove that d is pseudo-elementary. Expand the language L = { ., e } of' groups to L' = { ., e, + , *+O, 1, T , ~ where } + ,* are binary function symbols, 0 and 1 are constant symbols, and vij,1 5 i, j 5 n, are unary function symbols. Let d'

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be the class of all models 2l = (A, ., e, + ,*, 0,1, T , , ) for L’ satisfying the following properties (which are easily seen to be expressible as sentences in L’): (i) two elements ul, u2 of A are equal if and only if w i j ( u l ) =7rij(u2)for all i , j (hence elements of A “are” n X n matrices); (ii) the union F of the ranges of the 7rij forms a field with respect to + , * , O , and 1; (iii) the “matrix” e is the identity matrix; (iv) the operation * is matrix multiplication; and (v) the elements of A are precisely the n X n matrices over F which are invertible. Then d is the image of d ’ under the forgetful functor. 0 The class d in the above theorem is known not to be axiomatizable. (See SABBAGH [1969];the proof is given for n = 1, but it generalizes to arbitrary n.) It is an open problem whether the class in the next theorem is elementary or not. 4.6. THEOREM (Sabbagh). The class of primitive rings is closed under

ultraproducts. PROOF.For our purposes, the most convenient definition of a primitive ring is that R is primitive if and only if there is a maximal regular right ideal p in R such that ( p : R ) = (0) (see HERSTEIN [1968], p. 40). With this definition it is not hard to see that the class of primitive rings is pseudo-elementary. 0 We conclude this section with one other useful result about ultrapowers. 4.7. THEOREM. Let D be an ultrafilter over a set I . There is a natural

transformation d : I-* Il,, from the identity functor on A(L) to the ultrapower functor on A(L) such that for each ?I in A(L), d(?I) is an elementary embedding. Moreover, for any expansion. L‘ of L we have RL.Ld= d R L . L .

PROOF.For any 41 in A(L) let d ( ? i ) :?I-,Il,?I be defined by d ( a ) = iD where 6 : I+ 91 is the constant function with value a. That d(?I) is an elementary embedding follows from the fundamental theorem, 3.1, since

nD?ft=

.

cp [ d ( a , ) . . d ( a . ) ] e { i E

I I 91, = ?I I= cp[ a , . . . a n ] }E D

-3? i ! = c p [ a , . . . a ” .]

118

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(CH.A.3, 55

We leave to the reader the easy verification that d is a natural transformation (i.e. d ( 8 ) o f = II,(f)od(YI) for any f : 9 l j - B ) and that R L L d = d RL.L. 0

COMPACTNESS

5. An ultraproduct version of the Compactness Theorem The fundamental theorem implies that for any family (91, : i E I ) and any ultrafilter D, if each ?I, satisfies a set of first-order sentences 2, then nD?1, also satisfies 2. The content of the next theorem (which is an easy corollary of the fundamental theorem) is that for certain ultrafilters D, the ultraproduct n,?l, may satisfy a set of first-order properties s though no %, is a model of 2. 5.1. THEOREM. Let (?I, : i E I ) be an indexed family of models for L and let

2 be a set of sentences of L.

(i) I f for every sentence cp of 2, { i E I 1 ?I, I= cp} is cofinite, then for any non-principal ultrafilter D over I, n,91, is a model of 2. (ii) Ifforeveryfinitesubset {cp ,,..., cp,} o f 2 , { i E I l ? l , ~ c p l ~ . . . ~ cisp " } non-empty, then there exists an ultrafilter D over I such that II, 91, is a model of 2.

PROOF.(i) By Proposition 1.4, if D is non-principal, then { i E Z 1 ?I, I= cp} is in D, for every cp E 2, and therefore, by Corollary 3.2, n,?I, I= cp. (ii) For each finite subset F of 2, let I F = { i E I 1 ?I, I= F}; I F is non-empty by hypothesis. Notice that S = { I F 1 F is a finite subset of 2}has FIP since I F , n * * f l IF. 2 IG where G = 6. Thus by Corollary 1.2, S is contained in an ultrafilter, D, over I. Since I{,+)€ D, it follows from Corollary 3.2 that II,?l, I= cp for every cp E 2. 0

uy=,

As a corollary of the theorem we may obtain an algebraic proof of the Compactness Theorem discussed in the previous chapters.

5.2. COROLLARY. If 2 is a set of sentences of L such that every finite subset of 2 has a model, then 2 has a model.

PROOF.Let I be the set of all finite subsets of 2 and for each i E I, let ?Ii be a model of i. It is clear that the hypothesis of 5.l(ii) is satisfied, and

CH.

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AN U L T R A P R O D U a VERSION OF THE COMPACTNESS THEOREM

119

therefore there is an ultrafilter D over I such that nD?liis a model of

2

0

Most, if not all, of the algebraic applications of ultraproducts which are given in this and the next two sections can be proved without use of ultraproducts by using instead Corollary 5.2, the Compactness Theorem. Many of them were first proved in this way. However, it is the goal of this paper to show how ultraproducts may be used to give more “algebraic” proofs. Thus our proofs are more-or-less direct applications of Theorem 5.1 or Theorem 3.1 and even the appeal to these general theorems can be replaced in specific applications by a direct verification from the definition of ultraproduct of the desired properties (cf. the remark at the end of the proof of 3.4).

5.3. THEOREM. Let P be a infinite set of primes and for each p E P let F, be a field of characteristic p. If D is a non -principal ultrafilter over P, then n OF, is a field of characteristic zero. PROOF.Let 2 = {cp, 1 n 2 1) where cpn is the sentence “ n . 1 # 0”. For each n, {p E P 1 F, I= cp,} is cofinite since it is the set of primes in P not dividing n. Hence by 5.1 (i), IIDF, is a model of C i.e. nDF, has characteristic zero. 0 A class is finitely axiomatizable (in L) if it is of the form d ( C )where

2 is

a finite set of sentences of L; notice that in this case we may assume-by forming the conjunction of the sentences in 2 -that C consists of a single sentence. If .d= d({e}) is a finitely axiomatizable subclass of d ( L ) then the complement of ,d in A(L) is finitely axiomatizable-it is d ( { i6))and hence, by 3.3, closed under ultraproducts. Since by 5.3, the class of fields of non-zero characteristic is not closed under ultraproducts we obtain the following. 5.4. COROLLARY. The class of fields of characteristic zero is axiomatizable

but not finitely axiomatizable. Similarly we can prove the following. (For a proof and other examples of non-finite axiomatizability, see Chapter 5, 0 3 , of BELLand SLOMSON [ 19691.)

5.5. THEOREM (Tarski). The class of algebraically closed fields is axiomatizable but not finitely axiomatizable.

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66

5.6. THEOREM. If cp is a sentence of the language of fields which is true in

every field (resp. every algebraically closed field) of Characteristic zero, then there is a finite set of primes P, such that cp is true in every field (resp. every algebraically closed field) of characteristic p if pfZ P,. PROOF.Suppose not. Then there is an infinite set of primes P such that for each p E P there is a field (resp. algebraically closed field) F, which is a model of -I cp. By 5.3 and 3.2, if D is a non-principal ultrafilter over P, II& is a field (resp. algebraically closed field) of characteristic zero which is a model of i c p , a contradiction. 0 We close this section with a definition which plays an important role in a very deep result of Ax [I9681 that the theory of finite fields is decidable. Let 2 be the set of all sentences of the language of fields which are true in every finite field. An infinite model of 2 is called a pseudofinite field. (Ax (19681 gives a purely algebraic description of these fields.) The following immediate corollary of 5.l(i) shows how pseudofinite fields may be constructed as ultraproducts. 5.7. COROLLARY. Let (F,: n E w ) be a family of finite fields such that for each m , { n E w 1 cardinality of F. 5 m } is finite. Let D be a non-principal ultrafilter over w . Then n D F , is pseudofinite. 0

6. Embedding theorems

The following theorem is an abstraction of a, method frequently employed in proofs involving ultraproducts (cf. KEGEL and WEHRFRITZ [ 1973]), p. 66; also GRATZER [1968], Theorem 7, p. 261). A set {a,1 i E I } of substructures of B is called a local system for B if: (1) B = U { B , i E I } ; and (2) for every i, j E I there exists k E I such that B,U B, C €3., For example, the set of all finitely-generated substructures of B is a local system for ‘P.

I

6.1. THEOREM. Let {B, I i E I } be a local system for a model B E A(L). Suppose (?I,: i E I ) is an indexed family of members of A(L) such that for each i E I, there is an embedding (resp. homomorphism) of 8, into a,. Then there is an ultrafilter D on I and an embedding (resp. homomorphism) of 8 into &,a,.

CH A.3, $61

EMBEDDING THEOREMS

121

PROOF.Choose an embedding (resp. homomorphism) 6, : '23, + %, for each i E I. For each j E I, let S, = { i E I 1 B, C B,}.Then S = {S, 1 j E I}has the FIP since S,, r l rl S,, 3 Sk where k is such that B,,U * U B,. C B,. Hence by Corollary 1.2, S is contained in an ultrafilter D over I. For each j E I, there is an embedding (resp. homomorphism) v, : '8, -+ IID8,given by: g, ( b ) = bD where 6 ( i ) = [, ( b ) if i E S, and 6 ( i ) is arbitrary otherwise. Now by definition of a local system, '23 is the direct limit of the direct system consisting of the B,, i E I, and inclusion maps between them. The functions (7,l j E I} form a compatible family of maps with respect to this direct system and hence they induce a function 71 : '23 + UD%,which is easily seen to be an embedding (resp. homomorphism). 0 6.2. COROLLARY. If d C&(L) is closed under ultraproducts and i f every finitely-generated substructure of 8 E &(L) is embeddable in a member of d,then 8 is embeddable in a member of d. 0

A group G is called a linear group of degree n if it is isomorphic to a subgroup of G L ( n , F ) for some field F. The following is an immediate consequence of Corollary 6.2 and Theorem 4.5. 6.3. THEOREM (MALTEV[1940]). Let G be a group such that every finitely generated subgroup of G is a linear group of degree n. Then G is a linear group of degree n. 0

For the sake of simplicity of exposition and proof we shall assume for the remainder of this section thar our language L has a finite vocabulary, i.e. only a finite number of relation, function and constant symbols. If ?I is a model for L an equation over '?I is an expression of the form yo = f ( - y , . . - y n ) or R ( y , . * . y . ) where f (resp. R ) is an n-ary function (resp. relation) symbol and each yi is a variable or constant symbol of L or an element of A (or, more precisely, a new symbol c. representing an element a of A ) . An inequation ouer ?l is the negation of an equation over Yf . If Yf C 8 we say Yf is algebraically closed (resp. existentially closed) in 8 if any finite system of equations (resp. equations and inequations) over ?f which has a solution in '23 has a solution in Yf. We say ?I E d is algebraically closed (resp. existentially closed) in d if 91 is algebraically closed (resp. existentially closed) in every extension which is a member of d. (A synonym for existentially closed is existentially complete. For more on these notions see Chapter A.4.)

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6.4. THEOREM. Suppose 91 C 8 . Then ?S is algebraically closed (resp. existentially closed) in 'P if and only if there is an ultrafilter D and a commutative diagram

such that 7 is a homomorphism (resp. embedding). PROOF.For sufficiency, notice that 7 carries a solution of a finite system S of equations (resp. equations and inequations) with parameters from A into a solution of d(S)-i.e. S with parameter a replaced by d(a)-in II,%. If bg),. . ., b$" is a solution of d ( S ) in n,%, then by Theorem 3.1 there is a (non-empty) set X in D such that for each i E X , b(')(i), . . ., b'"'(i) is a solution of S in 3 . (Alternately, one may use the fact (4.7) that d is an elementary embedding: a fortiori d(91) is existentially closed in n, 71.) For necessity, we first note that without loss of generality we may assume that L has no function symbols (by replacing functions by their graphs). Let I be the set of all pairs (Fl,F2) where FI is a finite subset of A including the interpretations of all constant symbols of L, and Fz is a finite subset of B - A. Given i = ( F , ,F,) E I, where, say, F2 = { b l .. . b n } ,let S be the set of all equations (resp. equations and inequations) in x I .. . x, with constants from FI which are satisfied in 8 when we let x, = b, for j = 1,. . ., n. Since S is finite, S has a solution x I= a l ,.. .,x, = a, in 91. If 8 , denotes the substructure of 'P with universe F , U F2 (here we use the fact that L has no function symbols) then there is a homomorphism (resp. embedding) 4, : 'P, -91 such that 6, is the identity on F , and l , ( b , )= a, for j = 1 , . . .,n. Applying Theorem 6.1 to the local system (23,1 i E I}, we obtain the desired homomorphism (resp. embedding) 7 : 93- n,91. 0 In similar fashion one can give necessary and sufficient conditions for 9I to be an elementary substructure of 'P (see BELLand SLOMSON [1969], Chapter 8, § I ) . The following nice application of 6.4 is due to SABBAGH [unpublished] and BACSICH [ 19721. (Special cases were previously known; for example the case of group is due to NEUMANN [1952].) A model 9S E d is called simple in d if every non-constant morphism in d with domain 91 is on e-on e.

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EMBEDDING THEOREMS

6.5. THEOREM. Suppose d is closed under ultraproducts and suppose every member of d can be embedded in a simple member of d. If '21 is algebraically closed in d and non-trivial (i.e. has cardinality 2 2), then YI

is existentially closed in d and ?I is simple in d. PROOF.To prove the first part we must show that if 91 is a submodel of

23 E d,then ?I is existentially closed in %. By hypothesis, we may assume that 23 is simple. By 6.4 there is a commutative diagram n,91

where 7 is a homomorphism. Now 7 is not constant since 91 is non-trivial. Hence 7 is one-one and so by 6.4 91 is existentially closed in 23. To see that '?I is simple, consider a non-constant homomorphism f : ?I 6. Now ?I can be embedded in a simple model C and by 6.4 and 4.7, we have a commutative diagram

-

where 7 is a homomorphism. Since d ( C ) is one-one, Il,f 7 is nonconstant and therefore (since 6 is simple) one-one. Since the diagram commutes, we conclude that f is one-one. 0 0

We conclude this section with a result about embeddings of rings due to

A. Robinson and M . Rabin (see ROBINSON[1962]). We could derive the result from Theorem 6.1, but it is simpler to give a direct proof. A ring R (possibly without an identity) is called a prime ring if for all non-zero a and

p in R , aRP is non-zero.

6.7. THEOREM. Let d be a class of rings closed under ultraproducts. If R is a prime ring which is embeddable in a direct product of rings in d,then R is embeddable in a member of d,

PROOF.Suppose R C n,A,,where A, E d for all i E I. For each non-zero

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[CH.

A.3, 57

a in R let S, = { i E I l a ( i ) # 0). Since R is a prime ring, S = {S, I a # 0, a E R } has FIP (because S, f l S, 3 SUrP for some y E R ) . Thus by Corollary 1.2, S is contained in an ultrafilter D over I. Let A = UDAi. By hypothesis, A E d.Define a map 7 : R - A by q(a)= a D ;then g is an embedding since for each non-zero a in R, S, E D.

The following corollary is due to ROBINSON [ 19621 for t h e case of division rings, and AMITSUR [I9671 for the case of primitive rings. See AMITSUR [ 19671, HERSTEIN [ 19681, Chapter 7 or HiRscHELMmN [ 19721 for generalizations and an important application to Posner's Theorem. 6.8. COROLLARY. If R is a prime ring which is a subring of a direct product of division rings (resp. primitive rings), then R is a subring of a division ring (resp. primitive ring).

PROOF.The class of division rings is elementary, hence closed under ultraproducts. The class of primitive rings is pseudo-elementary, hence closed under ultraproducts (see Theorem 4.6). Some .more applicatibns of Theorem 6.7 to the study of polynomial identities and rational identities of division rings can be found in AMITSUR [I965 and 19661. 7. Bounds in polynomial ideals Throughout this section we shall be considering polynomial rings F [ X , . . .X.] over a field F, where n, is an arbitrary but fixed positive integer. A polynomial in F [ X , . . .X,] will be written as f = E Mc MXM where M ranges over all n-tuples ( m , . ..m,)of natural numbers and X M is an abbreviation for X ; " ' X ?. . . X ? . The degree of M, denoted S ( M ) , is C : = , m , ;and the degree of f is max{S(M)(cM#O}. Let (F,:i E I) be an indexed family of fields and let D be an ultrafilter over I . Let F denote the field IIDE and for c E n,F, let E denote cD. Let R denote the ring n,(F,[X,...X.]) . Now R is not a polynomial ring over F-it is not even Noetherian-but there is a canonical mapping p :F[X,...X,]+

defined as follows. If f p ( f ) = f D where f ( i ) =

=

R

XMCMXM E F [ X , . . . X.] and deg f = d, then dcM(i)XM.

M.A.3,

%7]

125

BOUNDS IN POLYNOMIAL IDEALS

--

An indexed family f = (fi: i E I) of polynomials fi E E [XI -X.] is called bounded if there exists m such that deg fi 5 m for all i E I. We leave to the reader the task of checking the following. 7.1. THEOREM. The function p is a well-defined embedding of rings whose

range is the subring of elements of R represented by bounded families. Moreover, for any f E PrX,.* X,,] and any Cil,. . ., 6,,E P, we have f(iil...ci.)= ii, where v ( i ) = f ( i ) ( a l ( i ) . . . a , ( i.) ) If f = (fi : i E I ) is a bounded family, let f denote the unique element of P [ X , . - . X . ] such that p ( ( f ) = fD. As a consequence of 7.1 and the fundamental theorem we obtain the following. 7.2. COROLLARY. Let f = (fi: i E Z), g'"= (g!'): i E Z), h'"= (h:'):i E I ) be bounded families of polynomials, for 1 = 1,. . ., r. Then

Moreover, for any 61,.. ., 6. in { i E I I f i ( a l ( i ) .. a,(i j) = 0) E D.

F,

f(dl.

* a & )

=6

if and only if

+

Let us consider some applications. 7.3. THEOREM (ROBINSON [ 1955a1). Given positive integers n and d there exists a positive integer m such that for any algebraically closed field F and any polynomials f , g'l),. . ., g'" in F I X I - .. X n ] of degree 5 d, if every common zero of g"), . . ., g'" in F is a zero of f , then f " belongs to (g"), . . ., g")), the ideal generated by g(I),. . ., g'".

PROOF.Note that the content of the theorem is that m can be chosen to depend only on n and d. Hilbert's Nullstellensatz, which we use in the proof, asserts that given f, g(l),. . ., g'" satisfying the hypotheses, there is some m -a priori depending on f , g'l),. . ., g'"-such that f " E (g'l),. . ., g(')).If the theorem is false, then for each i E Z = w there exists an algebraically closed field Fi and polynomials f,, g?, . . ., g?E E [ X I . . ex.] of degree 5 d such that f f fiZ (gY),...,gl')) but every zero of g { ' ) -* . g!" in F, is a zero of fi. (Note that we can assume that r does not depend on i since every ideal generated by polynomials of degree 5 d has a basis of cardinality I the dimension (over the field of coefficients) of the vector space of polynomials of degree 5 d.) Let D be a non-principal ultrafilter

126

[a. A.3, 97

EKLOFIULTRAPRODUCIX

over I. Since (fi : i E I ) and (g!'):i E I ) , 1 = 1 , . . ., r are bounded families we may consider f,g('),. . ., g'')E F[X, X.]. Every zero of g"), . . ., g'') in P is a zero of f by Corollary 7.2 since this is true for all i E I. Hence by the Nullstellensatz, f" E (g('),. . ., g!')) for some k E w. Then by 7.2 we have { i E Z 1 f f E (g!'), . . ., g?))} E D. But this contradicts the choice of fi, g!'),. . .,g ? ) and the tact that every element of D is infinite. 0

-

The following result, giving bounds for Hilbert's basis theorem, is proved in a similar fashion. (See SEIDENBERG [1971] for a purely algebraic proof.) 7.4. THEOREM. Given positive integers n and d there exists m such that for

-

any field F, any strictly ascending chain of ideals in F [X I * * X.] which are generated by polynomials of degree 5 d is of length 5 m.

One can also employ these methods in order to obtain bounds for the number of squares required to represent a positive definite rational functional over an ordered field as a sum of squares (see, for example, ROBINSON [1973b]). The following result about bounds requires a different mode of proof (see ROBINSON [1973a] for a discussion of the difference). It was first proved by KONIG [1903] and HERMANN [1926] using complex computational methods. The following proof is due to ROBINSON [1973a].

7.5. THEOREM. Given positive integers n and d there exists m such that for any field F and any polynomials f, g(l),. . ., g'" in F I X I . . oxn]of degree 5 d, if f E ( g " ) , . . ., g'')), then there exist polynomials h('),. . ., h'" in F I X I . . a x n ] of degree 5 m such that f = C:=,h'"g"'. PROOF.By standard considerations of linear algebra it suffices to prove the theorem for algebraically closed fields. Suppose the theorem is false. Then for every i E I = o there exists an algebraically closed field F, and polynomials f , , g?, . . . ,g ? ) E F, [X, . . Xn] of degree Id such that fi E ( g ! ' ) ,. . ., g!')), but there are no polynomials h (,1 ) , . . ., h,") of degree 5 i (1) (1) such that f , = C;=, h , g , . Let D be a non-principal ultrafilter over I and consider g"), . . ., g''' E F[X,. * X.]. Let G denote (f'), . . .,#(')), the ideal generated by g(",. . ., g(')in P [ X , . . - X n ] If . we can prove that f E G then we will be done since then there will exist 6"), . . ., i'"in P [ X , . * e x n ] such that f = C;=, i(')g''), and this, by 7.2, will contradict the choice of the (1) (r) f , , g , ,..., g , . Since the case G = F I X l . . . X n ] is trivial, we may assume that G =

-

5

-

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127

. . - r l Qk where the Qi's are proper primary ideals (cf. ZARISKI and SAMUEL [1958], Chapter IV, 04). So it suffices to prove that f € Q, for j = 1,. . ., k. Consider a fixed Qi = Q. By a change of coordinates we may assume that Q C ( X l ,.. . , X n ) . ( I f ( a l , ..., an) is a zero of Q in F, let Y , = X , - a, for v = 1 , . . ., n.) Let P denote (XI,. . .,X.). By the Krull Q1n

Intersection Theorem (ZARISKI and SAMUEL [ 19581, Chapter IV, Theorem 12'), it suffices to prove that f E Q + P"' for every m 2 0. Now since fi E ( g ! ' ) ,. . ., g!')), there exist polynomials p i ' ) . . -pi" such that 0) class of ( p i : i E Z) in R. Then f D = fi r= C (rl1 =) l( 1p()1 i) g (i1 ). Let p (D1 ) be (the 1) (1) ~ ~ g D , i.e. = p ( ~f ) = ~ p; = , pp~, ( g ( ' ) ) .Fix m >o. Let qD be the class of (0 i E Z) where q i = C b ( M ) s , ~ M ( i ) Xif M pl"= C M c M ( i ) X M Notice . that qD belongs to the range of p ; say q x ) = p (q")). Then

The left-hand side clearly belongs to the range of p ; thus the right-hand side shows that it belongs to p ( P m ) .We conclude that f E Q + P". 0 The following result is due to HERMANN [1926] (see also SEIDENBERG [1974]). It would be of interest to have an ultraproduct proof of this result in the spirit of the above, avoiding the computational methods of Hermann. None is known at present. 7.6. THEOREM. Given positive integers n and d there exists m such that for

-

any ideal J of F I X l * * X n ] which has a basis consisting of polynomials of degree 5 d, if J is not prime, then there exist polynomials f , g of degree 5 m such that f g E J but f $Z J and g J. For an application of the methods of this section and Section 5 to the problem of resolutions of singularities in algebraic geometry, see EKLOF [ 19691.

SATURATION

8. Ultraproducts which are wl-saturated In Sections 5-7 we used ultraproducts to construct models which satisfied given sets 2 of first-order properties. Now we want to study a property of ultraproducts (with respect to certain ultrafilters) called wl-saturation

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[CH.A.3, 08

which is not first-order but which is model-theoretic in character and is very useful in studying first-order properties of models. Recall from Chapter A.2 that if '21 is a model for L and X is a subset of A, then Lx denotes the language obtained by adding a constant symbol c. for each a E X ; and 91, = (9, a).=, is the model for L x obtained by interpreting the new constant c. by the element a. Let f ( u ) be a set of formulas y(u) of L, which have u as the only free variable; f ( u ) is called a type in 91, if tor every finite subset { y o ( u ) ,. . ., ym(u)}of f ( u ) there exists a E A such that 91, C y , [ a] for i = 0,. . ., m. We say f is realized in 91, if there exists a E A such that 41, C y [ a ] for all y(u) in T ( u ) .A model '21 for L is called w,-saturated if for every countable subset X of A, every type in ?I, is realized in 2'1., In order to clarify the definition let us begin with two important examples. A linearly ordered set ?I is called an q,-set (HAUSDORFF [1914]) if whenever B ,and B 2 are subsets of A of cardinality < w, such that B ,< B 2 (i.e. for all 6 , E B , , b2 E B2, we have 6 , < 6 2 ) , then there exists a E A such that B , < a < B z (i.e. for all 6 , E B,,62E B2 we have b , < a < 62). Obviously an qo-set is just a densely ordered set without first or last element. 8.1. LEMMA. If 41 is an qO-setwhich is w,-saturated, then 91 is a n 7,-set.

PROOF.Suppose B ,and B 2 are countable subsets of A such that B ,< B2. Let X = B ,U B, and f ( u ) be the set of all formulas of Lx of the form

cb, < u < cb where b , E B , and 62E B 2 . Since Vl is an qtr-set, f ( u ) is a type in 41., Therefore f ( n ) is realized in 41, by some a E A..It is then clear that B l < a
uncountable. PROOF.Suppose to the contrary that A is countable. Let f ( u ) be the set of all formulas of La of the form U#

ca

CH. A.3, 581

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129

where a E A. Since A is infinite, T ( v )is a type in ?IA and hence is realized in 21A, which is impossible. 0 We are going to prove that ultraproducts over the right kind of ultrafilter are wl-saturated; for this we need a new class of ultrafilters. An ultrafilter D over I is called wl-incomplete if there exist pairwise disjoint subsets Y. of I such that Y,, = I but for all n E w, Y n eD.An wl-incomplete ultrafilter is clearly non-principal since for a principal ultrafilter there exists a E I such that Y E D if and only if a E Y. The converse holds if I is countable.

u,,,

8.3. LEMMA.Every non-principal ultrafilter on a countable set I is wIincomplete.

PROOF.Say I = {a,,: n E w } . Let Y, = { a " } . Then Y , e D since D is non-principal, but Y. = I. 0

u,

It3s consistent with the usual axioms of set theory (ZFC) to assume that 8.3 holds for sets I of any infinite cardinality. However it is an open problem of set theory whether 8.3 can be proved (from ZFC) to hold for all infinite sets I. It is known that if 8.3 fails for I, then I is extraordinarily large. The cardinality of the smallest such I is a measurable cardinal, as discussed in the anonymous appendix to Chapter B.3. 8.4. LEMMA.For any infinite set I there is on of-incompleteultrafilter over I .

u,,,

PROOF.Since I is infinite we can write I = Y , where the Y.'s are pairwise disjoint and infinite. Let S = {I- Y.: n E w } . Then S has FIP so by Corollary 1.2, S is contained in an ultrafilter D which is w,-incomplete by construction. 0

8.5. THEOREM. Let L be a countable language and let D be an wlincomplete ultrafilter over I. For any family ('?Ii: i E I ) of models for L, the ultraproduct fl, 21i is wl-saturated. PROOF.Let X be a countable subset of nDAiand T ( u )a type in ( b ' ? I i ) x . Say f ( v ) = { y , ( v ) : n E w } . (Notice that f is countable since L is countable.) Let y,,, n € w, be subsets of I such that Y,,e D but u , , , , Y , = I. Since f ( o ) is a type in nDaithere exists for every m E o an element fb"' of nDAisuch that n,%, t= -y.[f'D"Q for n = 1 , . . .. rn. Now define g E n1Ai

130

EKLOF / ULTRAPRODUCTS

[CH.

A.3, 58

by g ( i ) = f""'(i) if i E Y,,,. We claim that nD?Ii C y,,,[gD] for all m E u.By Theorem 3.1 it suffices to prove that {i E I I Yli t= y m [ g ( i ) ] }= Z,,, is in D. But 2, contains I - urii Y, = Y,,,) and hence is in D since Yn6!D for all n. 0

nr,:(I-

A first-order theory T is said to be model-complete if whenever ?( and '23 are models of T such that ?I is a submodel of '23, then 8 is an elementary submodel of '23. For more on model-completeness see Chapter A.4. Here we simply want to show how ultraproducts may be used to prove that a theory is model-complete.

8.6. THEOREM (Robinson). The theory of real-closed fields is modelcomplete. PROOF.We shall assume the continuum hypothesis (CH). Suppose R , C R 2 are real-closed fields. Let q ( u , * u.) be a formula of the language of fields and let a l ,. . ., a. E R I such that R I C cp [al* . - a , ] . We wish to prove that R 2 C c p [ a l . . . a , ]. We claim that we may assume that R I and R 2 are countable. By the Lowenheim-Skolem theorem (see Chapter A.2) there is a countable elementary submodel R I of R l containing a l ,. . ., an,and there is a countable elementary submodel R :of R z containing R I. Thus we have countable real-closed fields R I CR:, R It= cp [ a l , .. ., a n ] ,and we wish to prove that R;t= cp [ a l , .. .,a,].This proves t h e claim. Let D be a non-principal ultrafilter on a countable set I. By Theorem 4.7 there are elementary embeddings d ( R , ) : R , + nDR, for v = 1,2. By 8.3 and 8.5, nDR, is w,-saturated and by 8.1 n D R , is an qI-set (with respect to the unique ordering on a real closed field) for v = 1,2. By a result of ERDOS,GILLMAN and HENRIKSEN [I9551 any two real closed fields of cardinality N1which are ql-sets are isomorphic, in fact any isomorphism of countable subfields extends to an isomorphism of the real-closed fields. Now by Lemma 8.2, Card(n,R,) 2 N,. But Card(n,R,) 5 Card(U,R,) = 2*(1=N1.Hence Card(nDR,) = NI for v = 1,2. Therefore by the result cited above, if e : R I + R 2 is the inclusion map, there is an isomorphism 4 : nDRI + nDRz such that

commutes. Hence

CH.

A.3, 581

131

ULTRAPRODUCT'S WHICH ARE UI-SATURATED

R II= cp [ a l .. . a,] =s ~ D I I=Rcp [ d ( R l ) ( a l ). .. d ( RI)(a.)]

e ndzk cp [ 4 ( d ( R l ) ( a l ) ) . 4 ( d ( R l ) ( a m ) ) ] e nDR21=cp [ d ( R z ) ( a l ) . . . d ( R 2 ) ( a . ) ]

e RZI=cp[ a l . - . a . ] . 0 (Although we assumed CH in the proof if follows from general logical considerations-which we shall not give here- that the result actually holds without this assumption.) Let us take the opportunity to mention that 8.6 has important algebraic consequences; in fact ROBINSON [ 1955bI (see also ROBINSON[ 1973bl) showed how the solution to Hilbert's seventeenth problem (solved originally by Artin) may be derived easily from Theorem 8.6. See 2.3 in Chapter A.4. 8.7. COROLLARY (Tarski). The theory of real-closed fields is complete i.e. any two real-closed fields are elementarily equivalent. PROOF.This follows immediately from 8.6 and the fact that any real-closed field contains a copy of the real closure of the rationals (cf. Theorem 6 of 1.7 in Chapter A.4). 0

In similar fashion we may prove the following result, using the wellknown theorem of Steinetz that any two algebraically closed fields of the same characteristic and the same uncountable cardinality are isomorphic. 8.8. THEOREM (Tarski-Robinson). Let p be 0 or prime. The theory of algebraically closed fields of characteristic p is complete and modelcomplete.

The continuum hypothesis is not needed in the proof of 8.8 because of the following result (see BELLand SLOMSON [1969], p. 130, for a proof). 8.9. THEOREM. If ( A i : i E I ) is a family of infinite sets and D is any wl-incomplete ultrafilter over I, then the cardinality of nDAiis at least 2"o.

We close with an interesting application of the methods of this and Section 5. The following is a special case of a theorem of Ax [1968]. 8.10. THEOREM. Let F be an algebraically closed field and let

VI : F" ---* F"

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EKLOFI ULTRAPRODUCTS

[CH.A.3, 59

be a polynomial map, i.e. *(x l . . . x . ) = ( ~ l ( x l . . . x , )...,I(I,( , xI...xn)) is injective, then is where .,,.,,~!,t are elements of F [ X , . * . X,,]. If surjective.

*

PROOF.For a fixed n and m and fixed degrees for t,hl,. . ., $,,, the theorem is expressible as a first-order sentence in the language of fields. Hence by Theorem 8.8 it suffices to prove the theorem for one algebraically closed field of each characteristic. For characteristic p, a prime, the theorem is easily seen to be true by a simple counting argument for F,, the union of all the finite fields of characteristic p. The theorem is true for n&, where D is a non-principal ultrafilter over P, the set of primes, by Corollary 3.2, and by Theorem 5.3, rIDFphas characteristic zero. 0 9. Ultraproducts of valued fields

One of the most important applicatiocs of ultraproducts to algebra is the work of Ax and KOCHEN[1965a, 1965b and 19661 and ERSHOV [1965] on Artin's Conjecture. The scope of this article dictates that we confine ourselves to giving an introduction to this work, sketching very briefly the role which ultraproducts play in the proof. We refer those whose appetite is whetted by this introduction to the original papers or to one of the many excellent expositions (e.g. CHANG and KEISLER[1973], KOCHEN[1975] and ROBINSON [1969]). See also the discussion in 2.4 of Chapter A.4. Artin's Conjecture states that the following property holds for all n and d when F = Q,, the field of p-adic numbers. (An.d)

Every homogeneous polynomial f E F I X I . . . Xnl of degree d such that n > d Z has a non-trivial zero in F.

The conjecture is is inspired by the similarities between Q, and Z,((t)), the field of formal power series over 2 , the field of order p. Lang proved that 2,( ( t ) )satisfies the property A K dfor all n and d. Artin's Conjecture in fact has been shown to be false for some p by TERJANIAN [ 19661. However, Ax-Kochen and Ershov proved the following. 9.1. THEOREM. For any positive integers n and d there is a finite set of primes P,,(n,d ) such that for every prime p !Z Po(n,d ) , A,,.., holds whea F = Q,.

At the heart of the proof of 9.1 is the following result which also gives a precise formulation of the intuitive analogy between Q, and Z , ( ( t ) ) .

CH. A.3,

091

ULTRAPRODUm OF VALUED FIELDS

133

9.2. THEOREM. For any non-principal ultrafilter D over P, the set of all primes, the ultraproducts n,Q, and n D Z p ( ( t ) ) are isomorphic fields. We leave to the reader the easy proof of 9.1 from 9.2. (Hint: use Corollary 3.2, Corollary 1.5, Lang's result and the fact that for fixed n and d, property An.d is a first-order statement.) As in the case of 8.6 and 8.8, Theorem 9.2 is proved by using algebraic properties possessed by the ultraproducts which arise from their being wl-saturated. We shall not give t h e proof, which is considerably more sophisticated than those of 8.6 and 8.8, but we d o want to indicate the key role played by wl-saturation. In fact Ax and Kochen prove that n D Q p and n D z p ( ( t ) ) are isomorphic as valued fields. Define a valued field (with cross-section) to be a model

where (F,+, - , O , 1) is a field; (A, ., 1,s) is an ordered abelian group; 1 1 : F - (0)- A is surjective and a valuation (i.e. 1 x y 1 = 1 x 1 I y 1, I x + y 15 max{ 1 x 1, 1 y I}); and 1 x 1 = x for all x E A. F is said to be w-pseudocomplete if for any sequence {a. 1 n E w } of elements of F such that (*)

1 a,

- a.

1 = I a n +-, a. I

whenever n < m < w, there exists a E F such that

1 a - a. 1 = 1 an+!- a. 1 for all n E w. The crucial result is then t h e following.

9.3. THEOREM. If F is an wl-saturated valued field, then F is w-pseudocomplete. PROOF.If X = { a . 1 n E w } is a sequence in F satisfying (*)-such a sequence is called w-pseudo-Cauchy -let T ( u )be the set of all formulas y.(u):

1 u - anI = I

-

an

I

for all n E w. T ( u ) is a subset of Lx, where L is the language of valued fields. Now T ( v ) is a type in Fx since for any m E w

Fx I= y. [a,+,] for n = 0 , . . ., m. Therefore there exists a E F such that

134

EKLOF / ULTRAPRODUCIS

[a. A.3, 010

FX t= y. [ a ]

for all n E w. (We say a is a pseudo-limit of X.) 0

10. Saturated models In the short space left to us we define the K-saturated and saturated models, which generalize the union of wl-saturated models, and mention without proof some important facts about these models. For more details and proofs see CHANGand KEISLER [1973], CHANG [1973] or BELLand SLOMSON [ 19691. Let K be an infinite cardinal. A model 91 for L is called saturated if for every subset X of 91 of cardinality < K , every type in 2' 1, is realized in ?Ix. l?I is called saturated if l?I is K-saturated where K is the cardinality of A. As in Section 8 we can prove: 10.1. LEMMA. (i) If 91 is an infinite model which is K-saturated, then 91 has cardinality 2 K . (ii) If 91 is a n qo-set which is we-saturated, then 91 is a n q=-set.

Models which are K -saturated can be obtained as ultraproducts, though the proof is more difficult than that of Theorem 8.5. The following theorem was first proved by KEISLER [1964] using GC H and by KUNEN[1972] without GCH. 10.2. THEOREM. Let L be a language of cardinality 5 K and let I be a set of power K . Then there is a n ultrafilter D on I such that for any family (%,: i E I ) of models for L, the ultraproduct nDYliis K+-Saturated.

(The ultrafilters with the property given by the above theorem are called '-good ultrafilters and were introduced by KEISLER [ 19641.) As a corollary we obtain the existence of saturated models. (The hypothesis of G C H at K is essential.) K

10.3. COROLLARY. Let K be a cardinal such that 2" = K +. If 91 is a model for L of cardinality 5 K (where L has cardinality IK ) , then there is a n elementary extension iI3 of 91 of cardinality K + which is saturated.

PROOF.Let I and D be as in Theorem 10.2. Then

iI3=nn,'21is

K+-

REFERENCES

135

saturated. By Theorem 4.7, 23 is an elementary extension of ?I. Moreover K + 5 Card(B) 5 2" by Lemma 10.1 and the definition of ultrapowers, so by hypothesis, Card(B)= K + and hence 23 is saturated. A key property of saturated models is the following, due to Vaught (see

MORLEYand VAUGHT [1962]). It explains the motivation behind the proofs of 8.6 and 8.8.

10.4. THEOREM. Elementarily equivalent saturated models of the same

cardinality are isomorphic. The following immediate consequence of 10.2 and 10.4 was also proved directly without GCH by SHELAH [1971]. 10.5. COROLLARY (GCH). Let 91 and 23 be models for the same language L. The following are equivalent: (1) 91 is elementarily equivalent to 23. (2) There is an ultrafilter D on a set I such that nD91 is isomorphic to

nDB.

Thus we see that elementary equivalence is an algebraic notion.

References AMITSUR, S. [ 19651 Generalized polynomial identities and pivotal monomials, Trans. A m . Marh. SOC., 114, 210-226. [I9661 Rational identities and applications to algebra and geometry, J. Algebra, 3, 304-359. [ 19671 Prime rings having polynomial identities with arbitrary coefficients, Proc. London Math., 111, 17, 470-486. A x , J., [I9681 The elementary theory of finite fields, Ann. of Marh., 88, 239-271. Ax, J., and S. KOCHEN [1965a] Diophantine problems over local fields 1, A m . J. Math., 87, 605-630. [1965b] Diophantine problems over local fields 11: A complete set of axioms for p-adic number theory, A m . J. Marh., 87, 631-648. [I9661 Diophantine problems over local fields 111: Decidable fields, Ann. of Marh., 83, 437-456. BACSICH, P.D. [ 19721 Cofinal simplicity and algebraic closedness. Algebra Unioersalis, 2, 354-360. BELL,J.L. and A . B . SLOMSON [ 19691 Models and Ulrraproducts, an introduction (North-Holland, Amsterdam).

136

EKLOF/ ULTRAPRODUIXS

CHANG, C.C. [I9671 Ultraproducts and other methods of constructing models, in: Sets, Models and Recursion Theory, edited by J.N. Crossley (North-Holland, Amsterdam) pp. 85-12], [ 19731 What’s so special about saturated models, in: Studies in Model Theory, edited by M. Morley, MAA Studies in Mathematics, Vol. 8 (Math. Assoc. Am., Buffalo, NY) pp. 59-65. CHANG, C.C. and H.J. KEISLER [ 19731 Model Theory (North-Holland, Amsterdam). EKLOF,P. [ 19691 Resolutions of singularities in prime characteristic for almost all primes, Trans. Am. Math. Soc.. 146, 429-438. ERDOS,P.,L. G I L L M Aand N M. HENRIKSEN [ 19.551 An isomorphism theorem for real closed fields, Ann. ofMarh., Ser. 2,61,542-554. Y. ERSHOV, [I9651 On the elementary theory of maximal normed fields, Dokl. Akad. Nauk SSSR, 165. [English translation in Sou. Marh. Dokl., 1390-1393.1 GRATZEK, G. [ 19681 Uniuersal Algebra (Van Nostrand, New York). HAUSDORFF, F. [ 19141 Grundzuge derhfengenlehre (Leipzig). [Reprinted by Chelsea, New York, 1955.1 H E R M A NG. N, [ 19261 Die Frage der endlich vielen Schritte in der Theorie der Polynomideale, Math. Ann., 95, 736788. HFKSTEIN, I.W. [ 19681 Noncommutative Rings, Carus Mathematical Monographs, No. 15 (Math. Assoc. Am., Buffalo, NY). HIRSCHELMANN, A. 119721 An application of ultra-products to prime rings with polynomial identities, in: Conference in Marhemarical Logic -London ’70,Lecture Notes in Mathematics, Vol. 255 (Springer, Berlin) pp. 145-148. KEGEL, O.H. and B.A.F. WEHRFKITZ I19731 Locally Finite Groups (North-Holland, Amsterdam). KEISLER.H.J. [I9641 Good ideals in fields of sets, Ann. of Math., 79, 338-359. [ 1%5] A survey of ultraproducts, in: Logic, Methodology and Philosophy of Science, edited by Y. Bar-Hillel (North-Holland, Amsterdam) pp. 112-126. KOCHEN,S. [I9611 Ultraproducts in the theory of models, Ann. of Math., Ser. 2, 74, 221-261. [197S] The model theory of local fields, in: Logic Conference Kiel 1974, Lecture Notes in Mathematics, Vol. 499 (Springer, Berlin) pp. 384-425. KONIG, J. [ 1903) Einleitung in die allgemeine Theorie der algebraischen Grossen (Teubner, Leipzig). KUNEN, K. [I9721 Ultrafilters and independent sets, Trans. Am. Math. Soc., 172, 299-306. LANG.S. 119711 Algebra (Addison-Wesley, Reading, MA, revised printing). LoS, J. [ 195.51 Ouelques remarques, theoremes et problemes sur les classes definissables d’algebres, in: Mathematical Interpretation of Formal Systems (North-Holland, Amsterdam) pp. 98-113.

REFERENCES

137

MAL'CEV,A.I. [1940] On faithful representations of infinite groups of matrices Mar. Sb. 8, 405-422. [English translation in A m . Marh. Soc. Transl. (2). 2 (1956) 1-21.] NEUMANN, B.H. [I9521 A note on algebraically closed groups, J. London Math. Soc. 22, 247-249. ROBINSON, A. [ 1975al 711e'orie Me'tamarhe'marique des Idiaux, Collection de Logique MathCmatique, Ser. A (Paris-Louvain). [ 1955bI On ordered fields and definite functions, Marh. Ann.. 130, 257-271. [I9621 A note on embedding problems, Fund. Math., 50, 455-461. [1969] Problems and methods of model theory, in: Aspecrs of Marhemarical Logic, C.I.M.E. 1968, 111 Ciclo (Edizioni Cremonese, Rome). [ 1973al On bounds in the theory of polynomials ideals, in: Selecred Questions of Algebra and Logic : Mal'ceu memorial volume (Novosibirsk) pp. 245-252. [1973b] Model theory as a framework for algebra, in: Studies in Model Theory, edited by M. Morley, MAA Studies in Mathematics, Vol. 8 (Math. Assoc. Am., Buffalo, NY)pp. 134-157. SABBAGH, G. [1969] How not to characterize the multiplicative groups of fields, J. London Marh. Soc. (2), 1, 369-370. SEIDENBERG, A. [I9711 On the length of a Hilbert ascending chain, Roc. A m . Mar!. Soc.. 29,443-450. [I9741 Constructions in algebra, Trans. A m . Marh. Soc., 197, 273-313. S. SHELAH, [ 19711 Every two elementarily equivalent-models have isomorphic ultrapowers, Israel J. Math., 10, 224-233. TERJANIAN, G. [I9661 Un contre-example 21 une conjecture d'Artin, C.R.Acad. Sci. Paris, Str. A, 262, 612. ZARISKI, 0. and P. SAMUEL [ 19581 Commurariue Algebra (Van Nostrand, New York).