Trivial extensions satisfying certain valuation-like properties

Abstract

In this article, we study the possible transfer of the properties of being an almost valuation ring, a pseudo-valuation ring and a pseudo-almost valuation ring between a commutative ring A and $A⋉E$ (the trivial ring extension of A arising from an A-module E). Also, we extend the notion of an almost pseudo-valuation domain to the context of arbitrary commutative rings (possibly with nontrivial zero-divisors), and we also study the possible transfer of this notion between A and $A⋉E$. Our aim is to provide examples of new classes of commutative rings satisfying the above-mentioned properties.

Keywords:

• Almost valuation ring
• going-down ring
• pseudo-almost valuation ring
• pseudo-valuation ring
• trivial ring extension
• valuation ring

2010 Mathematics Subject Classification:

• Primary 13B99
• Secondary 13A15
• 13G05
• 13B21

Disclosure statement

No potential conflict of interest was reported by the authors.