Rings which Admit Elimination of Quantifiers

We say that a ring admits elimination of quantifiers, if in the language of rings, {0, 1, +, * }, the complete theory of R admits elimination of quantifiers. THEOREM 1. Let D be a division ring. Then D admits elimination of quantifiers if and only if D is an algebraically closed or finite field. A ring is prime if it satisfies the sentence: Vx Vy 3z (x = 0 v y = 0 v xzy X 0). THEOREM 2. If R is a prime ring with an infinite center and R admits elimination of quantifiers, then R is an algebraically closed field. Let si be the class of finite fields. Let 04 be the class of 2 X 2 matrix rings over a field with a prime number of elements. Let T be the class of rings of the form GF(p n)e GF(pk) such that either n = k or g.c.d. (n, k) = 1. Let 9 be the set of ordered pairs (f, Q) where Q is a finite set of primes and f: Q -3 si U A U (6 such that the characteristic of the ring f(q) is q. Finally, let V be the class of rings of the form E[qEfQf(q) for some (f, Q) in @. THEOREM 3. Let R be a finite ring without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R belongs to W. THEOREM 4. Let R be a ring with the descending chain condition of left ideals and without nonzero trivial ideals. Then R admits elimination of quantifiers if and only if R is an algebraically closed field or R belongs to W. In contrast to Theorems 2 and 4, we have THEOREM 5. If R is an atomless p-ring, then R is finite, commutative, has no nonzero trivial ideals and admits elimination of quantifiers, but is not prime and does not have the descending chain condition. We also generalize Theorems 1, 2 and 4 to alternative rings. ?0. Introduction. This is a paper in applied model theory. In it we begin to explore what the well-developed concept of quantifier-elimination means in the context of associative ring theory. (When we say that a structure has a property normally attributed to theories, we mean that the complete theory of that structure has that property.) In [17] Tarski proved that an algebraically closed field admits elimination of quantifiers. In [11] Macintyre proved what may be viewed as a converse to Tarski's theorem: An infinite field which admits elimination of quantifiers in the language of rings is an algebraically closed field. We emphasize that Macintyre's theorem is about elimination of quantifiers in the language of rings. It is well known [17] that the theory of real-closed fields admits elimination of quantifiers in the language of rings augmented with the auxiliary predicate > for order. Received December 3, 1976; revised February 20, 1977. ' Partially supported under National Research Council Grant A-3040 while a guest at Simon Fraser University, Burnaby, B.C. 92 i 1978, Association for Symbolic Logic This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:28:04 UTC All use subject to http://about.jstor.org/terms RINGS WHICH ADMIT ELIMINATION OF QUANTIFIERS 93 Several of the theorems in this paper may be viewed as an attempt to weaken the hypothesis of Macintyre's theorem from field to some more general ring theoretic assumption and still prove the same result. Thus in this paper we generalize Macintyre's theorem. In ?4 we characterize for the first time the class of finite rings without nonzero trivial ideals which admit elimination of quantifiers. Let Si be the class of finite fields. Let A be the class of 2 x 2 matrix rings over a field with a prime number of elements. Let W be the class of rings of the form GF(p fl) E GF(p k) such that either n = k or g.c.d. (n, k) = 1. Let 2 be the set of ordered pairs (f, Q) where Q is a finite set of primes and f: Q -* d U A U W such that the characteristic of the ring f(q) is q. Finally, let W be the class of rings of the form ?qeof(q) for some (f, Q) in 2. We prove that a finite ring without nonzero trivial ideals admits elimination of quantifiers if and only if it belongs to the class W. We begin the mathematics of this paper in ?2 where we prove a version of the theorem of Macintyre referred to above and prove an important lemma on relativization of quantifier elimination. In ?3 we prove that a prime ring with infinite center admits elimination of quantifiers if and only if it is an algebraically closed field. As a corollary we deduce that a prime ring of characteristic zero admits elimination of quantifiers if and only if it is an algebraically closed field. In ?5 we characterize division rings which admit elimination of quantifiers. We show that a division ring admits elimination of quantifiers if and only if it is a finite or algebraically closed field. In ?6 we show that a semiprime ring with the descending chain condition on right or left ideals admits elimination of quantifiers if and only if it is an algebraically closed field or a finite ring which admits elimination of quantifiers (and hence as is characterized in ?4). In ?7 we generalize several of the theorems of this paper to the class of alternative rings. We show that under hypothesis similar to those used above an alternative ring which admits elimination of quantifiers is associative. This generalizes previous results of the author [12] that the theory of a split Cayley-Dickson algebra over an algebraically closed field does not admit elimination of quantifiers in the language {O, +, * }. In ?8 we show that the theory of atomless p-rings (p a fixed prime) admits elimination of quantifiers. This result demonstrates the sharpness of the theorems in ?3 and ?6 since ap atomless p-ring has an infinite center and is semiprime but is not prime and does not have the descending chain condition on right or left ideals. ?1. Background. We have tried to make this paper accessible to someone familiar with model theory as contained in the first half of [14] and algebra as is frequently presented in a first-year graduate course. The books [8] and [10] are recommended as general references for the algebra contained in this paper. Most of our notation and definitions are standard. For emphasis we recall some of them. The symbol l= denotes satisfaction; the symbol F denotes This content downloaded from 207.46.13.156 on Sat, 10 Sep 2016 04:28:04 UTC All use subject to http://about.jstor.org/terms