ON SEMINORMAL INTEGRAL DOMAINS WITH TREED OVERRINGS

Abstract

The following result uses and generalizes a recent result of Ayache on integrally closed domains. Let R be a commutative integral domain with integral closure R0(inside the quotient field K of R) such that each overring of R (inside K) is a treed domain and there exists a finite maximal chain of rings going from R to R0. Then R is a seminormal domain if and only if, for each maximal ideal M of R, either RM is a pseudo-valuation domain or, for some positive integer n, there exists a finite maximal chain, of length n, of rings from RM to (RM)0 each step of which is (an integral minimal ring extension which is) either decomposed or inert. Examples are given in which the latter option holds where R is one-dimensional and Noetherian.

Keywords

• Minimal ring extension, FCP, integral domain, overring, treed domain, pseudo-valuation domain, seminormal, integral closure, i-domain