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 Σ 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 ϵ A is a Σ 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.