On $S$-Noetherian Lattices

TL;DR

This paper introduces and explores $S$-Noetherian lattices, generalizing Noetherian rings, and establishes key properties, including a Cohen-Kaplansky type theorem and primary decomposition uniqueness.

Contribution

It defines $S$-Noetherian lattices, proves their equivalence with $S$-Noetherian rings via ideal lattices, and develops foundational theorems including primary decomposition.

Findings

A ring is $S$-Noetherian iff its ideal lattice is $S_L$-Noetherian.

Every $S$-prime element of an $S$-Noetherian lattice is $S$-compact.

Existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices.

Abstract

In this paper, we define and study $S$-Noetherian lattices as a natural generalization of Noetherian rings. We prove that a ring $R$ is $S$-Noetherian if and only if its ideal lattice, $Id(R)$, is $S_L$-Noetherian. Furthermore, we establish a Cohen-Kaplansky type theorem for $S$-Noetherian lattices, showing that $L$ is $S$-Noetherian if and only if every $S$-prime element of $L$ is $S$-compact. Finally, we introduce the concept of $S$-primary elements-a generalization of primary elements in multiplicative lattices and demonstrate the existence and uniqueness of $S$-primary decomposition in $S$-Noetherian lattices.