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arXiv · 2604.26058

On $S$-Noetherian Lattices

Sachin Sarode·Chetan Patil·Vinayak Joshi
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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.

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Sachin Sarode, Chetan Patil, Vinayak Joshi. 2026-04-28. On $S$-Noetherian Lattices. https://arxiv.org/abs/2604.26058

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