ON SEMINORMAL INTEGRAL DOMAINS WITH TREED OVERRINGS

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International Electronic Journal of Algebra
Volume:15 Issue:15
ON SEMINORMAL INTEGRAL DOMAINS WITH TREED OVERRINGS

ON SEMINORMAL INTEGRAL DOMAINS WITH TREED OVERRINGS

Pages : 157-172

View : 39 | Download : 11

Publication Date : 2014-06-01

Article Type : Research Paper

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 R0insert ignore into journalissuearticles values(inside the quotient field K of R); such that each overring of R insert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values(RM);0 each step of which is insert ignore into journalissuearticles values(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

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