Finite maximal chains of commutative rings

Let R subset of S be a unital extension of commutative rings, with (R) over bar 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, ((R) over bar, S) is a normal pair, each intermediate ring of R subset of S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many (R) over bar -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.