When every finitely generated ideal is S-principal

TL;DR

This paper introduces and explores the properties of $S$-Bézout rings, a generalization of Bézout rings, including their characterizations, relationships with other classes, and behavior in various ring constructions.

Contribution

It defines $S$-Bézout rings, investigates their properties, and provides new characterizations and constructions within commutative ring theory.

Findings

$S$-Bézout rings generalize Bézout rings.

Characterizations of $S$-Bézout rings in various contexts.

Construction of new classes of $S$-Bézout rings.

Abstract

In this paper, we introduce the concept of $S$-B\'ezout ring, as a generalization of B\'ezout ring. We investigate the relationships between $S$-B\'ezout and other related classes of rings. We establish some characterizations of $S$-B\'ezout rings. We study this property in various contexts of commutative rings including direct product, localization, trivial ring extensions and amalgamation rings. Our results allow us to construct new original classes of $S$-B\'ezout rings subject to various ring theoretical properties. Furthermore, we introduce the notion of nonnil $S$-B\'ezout ring and establish some characterizations.