Almost valuation rings

Document Type: Research Paper

Authors

1 Department of‎ ‎Mathematics‎, ‎University‎ ‎of Kashan‎, ‎P.O‎. ‎Box 8731751167‎, ‎Kashan‎, ‎Iran.

2 Department of‎ ‎Mathematics‎, ‎University‎ ‎of Qom‎, ‎P.O‎. ‎Box 3716146611‎, ‎Qom‎, ‎Iran.

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.

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