Commutative Regular Rings with Integral Closure

First order conditions are given which are necessary for a commutative regular ring to have a prime integrally closed extension. If the ring is countable these conditions are also sufficient. In [8] an example was given of a commutative regular ring with no prime model extension to a commutative integrally closed regular ring. In this paper we give (in §2) first order conditions which are necessary for a commutative regular ring, R, to have a prime extension to an integrally closed regular ring. In the case that R is countable these conditions are also sufficient (§3). They are, however, not sufficient in the case that R is uncountable (§4). In [1] and [5] (see also [6] and [9]) it was shown that the theory K — of integrally closed commutative regular rings without minimal (4 0) idempotents is the model completion of the theory of commutative regular rings, KCR. In §5 we show that a model of KCR has a prime extension to a model of K-— (the theory of integrally closed commutative regular rings) if and only if it has a prime extension to a model of K—-. 1. Preliminaries. A commutative ring R with unit is called regular (in the sense of von Neumann) if R satisfies Vx3y (x2y = x). BR is the Boolean algebra of idempotents of R. The operations of BR ate e. U e. = e. + e. e.e, and e.ne, = eie2' ^r ls trie maximal ideal space of R. It is well known that if R is regular and commutative then SR is the Stone space of BR. In a natural way R is a ring of functions on SR. (R 3 a —» a(-) defined by a(s) = a + s in R/s, foi s e SR any maximal ideal of R. If R is regular then R/s is a field.) It will be intuitively helpful to think of commutative regular rings as rings of functions in this way. If R is a commutative regular Received by the editors July 14, 1974. AMS (MOS) subject classifications (1970). Primary 13L05, 13B20; Secondary 02H15.