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Keywords:
$S$-Noetherian ring; $S$-primary ideal; $S$-irreducible ideal; $S$-primary decomposition
$S$-Noetherian ring; $S$-primary ideal; $S$-irreducible ideal; $S$-primary decomposition
Summary:
Let $R$ be a commutative ring with identity, and let $S \subseteq R$ 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 $sy\in {\rm rad}(Q)$. Also, we say that an ideal of $R$ is $S$-primary decomposable or has an $S$-primary decomposition if it can be written as a finite intersection of $S$-primary ideals. First we provide an example of an $S$-Noetherian ring in which an ideal does not have a primary decomposition. Then our main aim 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.
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