Commutative Regular Rings without Prime Model Extensions

It is known that the theory K of commutative regular rings with identity has a model completion K . We show that there exists a countable model of K which has no prime extension to a model of K'. If K and K ate theories in a first order language L, then K is said to be a model completion of K if K extends K, every model of K can be embedded in a model of K , and for any model A of K and models B,, B, of K extending A, we have (ßi«)aEA = (B2, a) A, i.e. B. and B, are elementarily equivalent in a language which has constants for the elements of A. If a theory K has a model completion K , then the models of K can reasonably be regarded as the "algebraically closed" models of K; for example, the theory of algebraically closed fields is the model completion of the theory of fields. It was shown in [3] that the theory K of commutative regular rings with identity (formulated in the usual language L fot rings with identity) has a model completion. We recall that a commutative ring R with identity is said to be regular (in the sense of von Neumann) if for any element a £ R there exists b £ R such that a b = a. (A good reference is Lambek [2].) The model completion K is given by the following axioms: (i) the axioms of commutative regular rings with identity; (ii) an axiom stating that there are no minimal idempotents, i.e. Vx(x2 = x A x / 0 -♦ 3y(y2 = y A y^0Ay + xAyx = y)\ (iii) a set of axioms stating that every monic polynomial has a root. Received by the editors July 30, 1973 and, in revised form, December 21, 197 3AMS (MOS) subject classifications (1970). Primary 02H13, 02H15; Secondary 13L05, 13B99.