S-J-Ideals: A Study in Commutative and Noncommutative Rings

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-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-J-ideals. We demonstrate the equivalence of S-J-ideals and right S-J-ideals in commutative rings with identity and provide examples to illustrate the connections between right S-prime ideals and J-ideals.