On <i>S</i> -( <i>m</i> , <i>n</i> )-prime ideals and their extensions

Let R be a commutative ring, m and n positive integers, and S a multiplicatively closed subset of R. We introduce the notion of S-(m,n)-prime ideals, which generalizes the classical concept of (m,n)-prime ideals. We investigate their fundamental properties, establish several characterizations, and study their relationships with S-prime and (m,n)-prime ideals. Moreover, we examine the behavior of S-(m,n)-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.