On S-(m,n)-prime ideals and their extensions

Abstract

Let [Formula: see text] be a commutative ring, [Formula: see text] and [Formula: see text] positive integers, and [Formula: see text] a multiplicatively closed subset of [Formula: see text]. We introduce the notion of [Formula: see text]-[Formula: see text]-prime ideals, which generalizes the classical concept of [Formula: see text]-prime ideals. We investigate their fundamental properties, establish several characterizations, and study their relationships with [Formula: see text]-prime and [Formula: see text]-prime ideals. Moreover, we examine the behavior of [Formula: see text]-[Formula: see text]-prime ideals under various ring-theoretic constructions, including homomorphic images, localizations, direct products, trivial ring extensions, and amalgamated algebras. Examples are provided to illustrate the theory.