Let R be a commutative ring with identity, S ⊂ R be a multiplicative set and J be an ideal of R. In this paper, we introduce the concept of S-J-Noetherian rings, which generalizes both J-Noetherian rings and S-Noetherian rings. We study several properties and characterizations of this new class of rings. For instance, we prove Cohen's-type theorem for S-J-Noetherian rings. Among other results, we establish the existence of S-primary decomposition in S-J-Noetherian rings as a generalization of classical Lasker-Noether theorem.

The authors thank the referee for the careful review and insightful comments, which have significantly improved the manuscript. All suggestions have been incorporated. The authors also thank the editor for their consideration.