On S-valuation rings

Let [Formula: see text] be a commutative ring and [Formula: see text] be a multiplicative subset of [Formula: see text]. In this paper, we introduce and study an [Formula: see text]-version of valuation rings, that is called the notion of [Formula: see text]-valuation rings. Therefore, we say that [Formula: see text] is an [Formula: see text]-valuation ring if for all [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text] or [Formula: see text]. Moreover, we study the transfer of this property in trivial ring extensions and amalgamated algebras along an ideal. Finally, we explore the graded version of this concept in rings graded by a torsionless grading monoid.