ON BOOLEAN-ALGEBRAS AND INTEGRALLY CLOSED COMMUTATIVE REGULAR-RINGS

In this paper we consider properties, related to model-completeness, of the theory of integrally closed commutative regular rings. We obtain the main theorem claiming that in a Boolean algebra B, the truth of a prenex SIGMA(n)-formula whose parameters a(i) partition B, can be determined by finitely many conditions built from the first entry of Tarski invariant T(a(i))'s, n-characteristic D(n, a(i))'s and the quantities S(a(i), l) and S'(a(i), l) for l < n. Then we derive two important theorems. One claims that for any Boolean algebras A and B, an embedding of A into B preserving D(n, a) for all a is-an-element-of A is a SIGMA(n)-extension. The other claims that the theory of n-separable Boolean algebras admits elimination of quantifiers in a simple definitional extension of the language of Boolean algebras. Finally we translate these results into the language of commutative regular rings.