When every finitely generated ideal is S-principal

In this paper, we introduce the concept of $S$-Bézout ring, as a generalization of Bézout ring. We investigate the relationships between $S$-Bézout and other related classes of rings. We establish some characterizations of $S$-Bézout 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 $S$-Bézout rings subject to various ring theoretical properties. Furthermore, we introduce the notion of nonnil $S$-Bézout ring and establish some characterizations.