ALMOST VALUATION RINGS

Abstract

The aim of this paper is to generalize the notion of almost valuation domains to arbitrary commutative rings. Also, we consider re- lations between almost valuation rings and pseudo-almost valuation rings. We prove that the class of almost valuation rings is properly contained in the class of pseudo-almost valuation rings. Among the properties of almost valuation rings, we show that a quasilocal ring R with regular max- imal ideal M is a pseudo-almost valuation ring if and only if V = (M : M) is an almost valuation ring with maximal ideal RadV (M). Furthermore, we show that pseudo-almost valuation rings are precisely the pullbacks of almost valuation rings.