Almost valuation rings

The aim of this paper is to generalize theý ýnotion of almost valuation domains to arbitrary commutativeý ýringsý. ýAlsoý, ýwe consider relations between almost valuation rings ýand pseudo-almost valuation ringsý. ýWe prove that the class ofý ýalmost valuation rings is properly contained in the class ofý ýpseudo-almost valuation ringsý. ýAmong the properties of almostý ývaluation ringsý, ýwe show that a quasilocal ring R with regularý maximal ideal M is a pseudo-almost valuation ring if and onlyý ýif V=(Mý:ýM) is an almost valuation ring with maximal idealý ýRadV(M) ý. ýFurthermoreý, ýwe show that pseudo-almost valuationýýrings are precisely the pullbacks of almost valuation ringsý.