Isonoetherian power series rings

Abstract

ABSTRACTFacchini and Nazemian proved that a valuation domain is isonoetherian if and only if it is discrete of Krull dimension ≤2 and they showed that this cannot be generalized from the local case to the global case: the 2-dimensional generalized Dedekind domain ℤ+Xℚ[[X]] is not isonoetherian. Let D be an integral domain with quotient field K. We provide necessary and sufficient conditions on D and K, so that the ring D+XK[[X]] is isonoetherian. We deduce that if D is integrally closed, then D+XK[[X]] is isonoetherian if and only if D is a semi-local principal ideal domain.