On S-J-Noetherian Rings

TL;DR

This paper introduces and studies $S$-$J$-Noetherian rings, a new class generalizing existing Noetherian concepts, and proves key properties including Cohen's theorem and primary decomposition.

Contribution

It defines $S$-$J$-Noetherian rings, explores their properties, and establishes foundational theorems like Cohen's theorem and primary decomposition for this class.

Findings

Established Cohen's-type theorem for $S$-$J$-Noetherian rings.

Proved existence of $S$-primary decomposition in these rings.

Generalized classical results to a broader class of rings.

Abstract

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.