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 $a \in A$ 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.