The Set of Intermediate Rings as a Boolean Algebra

Abstract

Let R ⊂ S be an extension of integral domains, Supp(S/R) = {P ∈ Spec(R): (S/R) P ≠ 0} the support of the R-module S/R, and [R, S] the set of intermediate rings between R and S. We prove the following theorem: If there is a maximal chain of rings from R to S of length n and Supp(S/R) consists of n maximal ideals, then ([R, S], ·, ∩) is a boolean algebra with cardinality 2 n . An interesting example is exhibited to illustrate the scope of this theorem.