A Study of S-Primary Decompositions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915555052617728 |
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| author | Singh, Tushar Ansari, Ajim Uddin Kumar, Shiv Datt |
| author_facet | Singh, Tushar Ansari, Ajim Uddin Kumar, Shiv Datt |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_00922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A Study of S-Primary Decompositions Singh, Tushar Ansari, Ajim Uddin Kumar, Shiv Datt Commutative Algebra 13A15, 13B02, 13C05, 13E05 A.0 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. |
| title | A Study of S-Primary Decompositions |
| topic | Commutative Algebra 13A15, 13B02, 13C05, 13E05 A.0 |
| url | https://arxiv.org/abs/2401.00922 |