Almost valuation rings

The aim of this paper is to generalize the notion of almost valuation domains to arbitrary commutative rings. Also, we consider relations 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 maximal 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.