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 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 ‎${rm Rad}_V(M)$‎. ‎Furthermore‎, ‎we show that pseudo-almost valuation‎‎rings are precisely the pullbacks of almost valuation rings‎.