On S-J-Noetherian Rings

Let $R$ be a commutative ring with identity, $S\subseteq R$ 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 charaterizations 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.