ON SEMINORMAL INTEGRAL DOMAINS WITH TREED OVERRINGS

Year 2014, Volume: 15 Issue: 15, 157 - 172, 01.06.2014

David E. Dobbs

https://doi.org/10.24330/ieja.266245

Cited By: 1

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