Prime Quantifier Eliminable Rings

The main result of this paper is that a prime ring with a finite center which admits elimination of quantifiers must be finite. This answers a question raised by the author [10; p. 99] and completes the classification of prime rings which admit elimination of quantifiers. That is, a prime ring admits elimination of quantifiers if and only if it is either an algebraically closed field, a finite field, or a 2 x 2 matrix ring over a finite prime field. Two features of this proof merit mentioning in this introduction. In [11] Rose and Woodrow developed the concept of ultrahomogeneity as distinct from other concepts of homogeneity with the motivation that the concept was a natural one for algebraic structures and could prove useful in problems involving quantifier elimination. Ultrahomogeneity proved to be a natural tool in two parts of the proof of Theorem 1. It was used in the following way. Conditions were derived which forced a ring to be uniformly locally finite. Countable homogeneous models always exist. And by results in [11] any quantifier eliminable homogeneous elementarily equivalent ring had to be ultrahomogeneous. However the existence of a countable ultrahomogeneous model implies K0-categoricity. Theorems about K0-categorical rings could then be employed. The other feature we wish to point out is the application in the proof of Theorem 1 of Cherlin's result that an K0-categorical nil ring is nilpotent [3]. Cherlin's result answered question (2) of [1]. His proof is highly imaginative involving the construction of an infinite class of existential 6-types. Before proceeding to list essential definitions we will briefly review the context for this problem. Quantifier elimination is a well-developed concept in model theory. It is frequently used to derive decidability results and is an assumption frequently made (with change of language) in stability-like model theory. This paper thus seeks to discover what a well-known model theoretic concept means in a particular class of "known" algebraic objects. Further, this problem is developing a history of its own. In [13] Tarski proved that an algebraically closed field admits elimination of quantifiers in the language of ring theory. In [7] 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. In [10] Rose began to examine how strong the assumption of quantifier elimination in the language of rings was for various classes of rings such as division rings, semiprime rings with d.c.c, finite rings, etc. Subsequently, Macintyre, McKenna, and van den Dries [8] showed that if various fields were assumed to admit elimination of quantifiers in their natural languages (language of ordered fields, valued fields) then they were the fields in those languages already known to admit elimination of quantifiers. In [14] Wheeler proved some of the Macintyre-McKenna-van den Dries results from the viewpoint of examining universal theories which admit amalgamation. Several ideas from [14] are used in this paper.