Existence and uniqueness of S-primary decomposition in S-Noetherian modules

Abstract

Let R be a commutative ring with identity, $S\subseteq R$ be a multiplicative set, and M be an R-module. We say that a submodule N of M with $(N{:}_{R}M)\cap S=\varnothing$ has an S-primary decomposition if it can be written as a finite intersection of S-primary submodules of M. In this paper, first we provide an example of the S-Noetherian module in which a submodule does not have a primary decomposition. Then our main aim of this paper is to establish the existence and uniqueness of S-primary decomposition in S-Noetherian modules as an extension of a classical Lasker-Noether primary decomposition theorem for Noetherian modules.

KEYWORDS:

• S-irreducible submodule
• S-primary decomposition
• S-primary submodule

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B02
• 13C05
• 13C60
• 13E05

Acknowledgments

Authors sincerely thank the referee for thorough review and very useful comments and suggestions to improve the article.