A Study of S-primary Ideals in Commutative Semirings
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908741451907072 |
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| author | Mahato, Amaresh Das, Sampad Mandal, Manasi |
| author_facet | Mahato, Amaresh Das, Sampad Mandal, Manasi |
| contents | In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the $S$-version of the uniqueness theorem using localization, for $S$-$k$-primary decompositions.
Also we show that the $S$-radical of every $S$-primary ideal is a prime ideal of $R$. Moreover, we investigate the structure of $S$-primary ideals in principal ideal semidomain and prove that each such ideal can be expressed of the form, $I = (vp^n)$, $n\in \mathbf{N}$ and for some $p \in \mathbf P -\mathbf P_S$ and $v\in R$ such that $(v)\cap S\neq \varnothing $, where $\mathbf P$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S\subsetneq R$, the set $\mathbf P_S$ defined by $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2512_24800 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Study of S-primary Ideals in Commutative Semirings Mahato, Amaresh Das, Sampad Mandal, Manasi Commutative Algebra In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the $S$-version of the uniqueness theorem using localization, for $S$-$k$-primary decompositions. Also we show that the $S$-radical of every $S$-primary ideal is a prime ideal of $R$. Moreover, we investigate the structure of $S$-primary ideals in principal ideal semidomain and prove that each such ideal can be expressed of the form, $I = (vp^n)$, $n\in \mathbf{N}$ and for some $p \in \mathbf P -\mathbf P_S$ and $v\in R$ such that $(v)\cap S\neq \varnothing $, where $\mathbf P$ is the set of all irreducible (prime) elements of R and for a multiplicative subset $S\subsetneq R$, the set $\mathbf P_S$ defined by $\mathbf P_S=\{p\in \mathbf P : (p) \cap S \neq \varnothing \}$. |
| title | A Study of S-primary Ideals in Commutative Semirings |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2512.24800 |