  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.
