  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.
