Prime e.c. commutative rings in characteristic n ≥ 2

Abstract

Let CR denote the first-order theory of commutative rings with unity, formulated in the language L = 〈 +, •, 0, 1〉. Virtually everything that is known about existentially complete (e.c.) models of CR is contained in Cherlin's paper [2], where it is shown, in particular, that the e.c. models are not first-order axiomatizable. The purpose of this note is to show that, in analogy with the case of fields, there exists a unique prime e.c. model of CR in each characteristic n > 2. As a consequence we settle Problem 8 in the list of open questions at the end of Hodges' book Building models by games ([5], p. 278).By a “prime” e.c. model of characteristic n ≥ 2 we mean one that embeds in every e.c. model of characteristic n. (The embedding is not always elementary, since [2] not all e.c. models of characteristic n are elementarily equivalent.) The prime model is characterized by the fact that it is the union of a chain of finite subrings each of which is an amalgamation base for CR. In §1 we describe the finite amalgamation bases for CR and show that every finite model embeds in a finite amalgamation base; in §2 we use this information to obtain prime e.c. models and answer Hodges' question.Our results on prime e.c. models were obtained some years ago, during the fall term of 1982, while the author was a visitor at Wesleyan University. The author wishes to take this opportunity to thank the mathematics department at Wesleyan for its hospitality during that visit, and subsequent ones.