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.

Key Words:

• Almost Bézout rings
• Almost valuation rings
• Amalgamated algebra along an ideal
• Amalgamated duplication
• Trivial extensions

2010 Mathematics Subject Classification:

• 13F30
• 13F05

ACKNOWLEDGMENT

The authors would like to express their sincere thanks to the referee for his/her helpful suggestions and comments.

Notes

Communicated by F. Tartarone.