Finite maximal chains of commutative rings

Abstract

Let R ⊆ S be a unital extension of commutative rings, with [Formula: see text] the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, [Formula: see text] is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many [Formula: see text]-subalgebras of S. Each chain of rings from R to S is finite if dim (R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim (R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.