S-j-ideals: a study in commutative and noncommutative rings

In this paper, we introduce the concept of S- (Formula presented.) -ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of (Formula presented.) -ideals apply to S- (Formula presented.) -ideals and examine their characteristics in various ring constructions, such as homomorphic image rings, quotient rings, cartesian product rings, polynomial rings, power series rings, idealization rings, and amalgamation rings. In noncommutative rings, where S is an m-system, we define right S- (Formula presented.) -ideals. We demonstrate the equivalence of S- (Formula presented.) -ideals and right S- (Formula presented.) -ideals in commutative rings with identity and provide examples to illustrate the connections between right S-prime ideals and (Formula presented.) -ideals.