Isonoetherian power series rings II

Abstract

We study the isonoetherian and “ACC d on ideals” properties in power series rings, polynomial rings, rings of the form (where I is an ideal of a commutative unitary ring R), (where D is an integral domain with quotient field K), Nagata’s idealization ring and almost pseudo valuation domain. We point out three mistakes in two works. Let R be a quasi-local proper subring of an integral domain S such that R is integrally closed in S. We prove that is u-isonoetherian if and only if R satisfies ACC d on ideals, S is Noetherian and there exists a maximal ideal M of S that is a divided prime ideal of R and R/M is a rank one discrete valuation domain with quotient field S/M. We prove that the ring is isonoetherian if and only if R is a rank one discrete valuation domain with quotient field S. We deduce an example of a u-isonoetherian ring extension such that the ring is not isonoetherian.