n-absorbing ideal factorization in commutative rings
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arXiv
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| Format: | Preprint |
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2023
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| _version_ | 1866929246956421120 |
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| author | Choi, Hyun Seung |
| author_facet | Choi, Hyun Seung |
| contents | In this article, we show that Mori domains, pseudo-valuation domains, and $n$-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain $R$ is a Mori locally pseudo-valuation domain if and only if each proper ideal of $R$ is a finite product of 2-absorbing ideals of $R$. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form $A+XB[X]$ where $A$ is a subring of a commutative ring $B$ and $X$ is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of $n$-absorbing ideals for some $n\ge 2$ if and only if $A$ and $B$ are both Artinian reduced rings and the contraction map $\text{Spec}(B)\to\text{Spec}(A)$ is a bijection. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_05231 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | n-absorbing ideal factorization in commutative rings Choi, Hyun Seung Commutative Algebra In this article, we show that Mori domains, pseudo-valuation domains, and $n$-absorbing ideals, the three seemingly unrelated notions in commutative ring theory, are interconnected. In particular, we prove that an integral domain $R$ is a Mori locally pseudo-valuation domain if and only if each proper ideal of $R$ is a finite product of 2-absorbing ideals of $R$. Moreover, every ideal of a Mori locally almost pseudo-valuation domain can be written as a finite product of 3-absorbing ideals. To provide concrete examples of such rings, we study rings of the form $A+XB[X]$ where $A$ is a subring of a commutative ring $B$ and $X$ is indeterminate, which is of independent interest, and along with several characterization theorems, we prove that in such a ring, each proper ideal is a finite product of $n$-absorbing ideals for some $n\ge 2$ if and only if $A$ and $B$ are both Artinian reduced rings and the contraction map $\text{Spec}(B)\to\text{Spec}(A)$ is a bijection. A complete description of when an order of a quadratic number field is a locally pseudo valuation domain, a locally almost pseudo valuation domain or a locally conducive domain is given. |
| title | n-absorbing ideal factorization in commutative rings |
| topic | Commutative Algebra |
| url | https://arxiv.org/abs/2307.05231 |