A Study of S-Primary Decompositions
Tushar Singh, Ajim Uddin Ansari, Shiv Datt Kumar
TL;DR
The paper addresses extending primary decomposition from Noetherian rings to $S$-Noetherian rings via $S$-primary ideals. It builds a framework of $S$-primary decompositions, including $S$-irreducible and $S$-prime notions, and proves existence and two versions of uniqueness theorems for such decompositions. A key contribution is showing that every proper ideal disjoint from $S$ has a finite $S$-primary decomposition in $S$-Noetherian rings, along with analyses of minimal $S$-prime sets and localization behavior, complemented by a counterexample where an $S$-Noetherian ring is not Laskerian. The results generalize Lasker-Noether decomposition to $S$-localized settings, offering structural tools for ideal decompositions in generalized Noetherian contexts with potential geometric and algebraic applications.
Abstract
Let $R$ be a commutative ring with identity and $S \subseteq R$ be a multiplicative set. An ideal $Q$ of $R$ (disjoint from $S$) is said to be $S$-primary if there exists an $s\in S$ such that for all $x,y\in R$ with $xy\in Q$, we have $sx\in Q$ or $sy\in rad(Q)$. Also, we say that an ideal of $R$ is $S$-primary decomposable or has an $S$-primary decomposition if it can be written as finite intersection of $S$-primary ideals. In this paper, first we provide an example of $S$-Noetherian ring in which an ideal 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 rings as an extension of a historical theorem of Lasker-Noether.