Some factorization properties of idealization in commutative rings with zero divisors
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866917629979000832 |
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| author | Eftekhari, Sina Jafari, Sayyed Heidar Khorsandi, Mahdi Reza |
| author_facet | Eftekhari, Sina Jafari, Sayyed Heidar Khorsandi, Mahdi Reza |
| contents | We study some factorization properties of the idealization $R \mathop{(\! + \! )} M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R \mathop{(\! + \! )} M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R \mathop{(\! + \! )} M$ is a BFR. We also characterize the idealization rings which are UFRs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_09984 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Some factorization properties of idealization in commutative rings with zero divisors Eftekhari, Sina Jafari, Sayyed Heidar Khorsandi, Mahdi Reza Commutative Algebra 13F15, 13B99 We study some factorization properties of the idealization $R \mathop{(\! + \! )} M$ of a module $M$ in a commutative ring $R$ which is not necessarily a domain. We show that $R \mathop{(\! + \! )} M$ is ACCP if and only if $R$ is ACCP and $M$ satisfies ACC on its cyclic submodules. We give an example to show that the BF property is not necessarily preserved in idealization, and give some conditions under which $R \mathop{(\! + \! )} M$ is a BFR. We also characterize the idealization rings which are UFRs. |
| title | Some factorization properties of idealization in commutative rings with zero divisors |
| topic | Commutative Algebra 13F15, 13B99 |
| url | https://arxiv.org/abs/2106.09984 |