On Almost Valuation and Almost Bézout Rings

Abstract

In this paper we extend the notion of almost valuation and almost Bézout domains to arbitrary commutative rings, and we investigate the transfer of these properties to trivial ring extensions and amalgamated algebras along an ideal. Our aim is to provide new classes of commutative rings satisfying these properties. As an immediate consequence, we show the failure of Anderson–Zafrullah's theorem on the integral closure of an almost valuation domain beyond the context of integral domains.