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| Main Authors: | , , , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2405.05971 |
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| _version_ | 1866910441636102144 |
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| author | Uçar, Zeynep Yılmaz Ersoy, Bayram Ali Tekir, Ünsal Koç, Suat Onar, Serkan |
| author_facet | Uçar, Zeynep Yılmaz Ersoy, Bayram Ali Tekir, Ünsal Koç, Suat Onar, Serkan |
| contents | In this study, we aim to introduce the concept of classical 1-absorbing prime submodules of a nonzero unital module $M$ over a commutative ring $A$ with unity. A proper submodule $P$ of $M$ is said to be a classical 1-absorbing prime submodule, if for each $m\in M$ and nonunits $a,b,c\in A,$ $abcm\in P$ implies that $abm\in P$ or $cm\in P$. We give many examples and properties of classical 1-absorbing prime submodules. Also, we investiage the classical 1-absorbing prime submodules of tensor product $F\otimes M$ of a (faithfully) flat $A$-module $F$ and any $A$-module $M$. Furthermore, we determine classical prime, classical 1-absorbing prime and classical 2-absorbing submodules of amalgamated duplication $M\bowtie I$ of an $A$-module $M$ along an ideal $I$. Also, we characterize local rings $(A,\mathfrak{m})$ with $\mathfrak{m}^{2}=0$ in terms of classical 1-absorbing prime submodules. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_05971 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On classical 1-absorbing prime submodules Uçar, Zeynep Yılmaz Ersoy, Bayram Ali Tekir, Ünsal Koç, Suat Onar, Serkan Rings and Algebras 13A15, 13C05, 13C13 In this study, we aim to introduce the concept of classical 1-absorbing prime submodules of a nonzero unital module $M$ over a commutative ring $A$ with unity. A proper submodule $P$ of $M$ is said to be a classical 1-absorbing prime submodule, if for each $m\in M$ and nonunits $a,b,c\in A,$ $abcm\in P$ implies that $abm\in P$ or $cm\in P$. We give many examples and properties of classical 1-absorbing prime submodules. Also, we investiage the classical 1-absorbing prime submodules of tensor product $F\otimes M$ of a (faithfully) flat $A$-module $F$ and any $A$-module $M$. Furthermore, we determine classical prime, classical 1-absorbing prime and classical 2-absorbing submodules of amalgamated duplication $M\bowtie I$ of an $A$-module $M$ along an ideal $I$. Also, we characterize local rings $(A,\mathfrak{m})$ with $\mathfrak{m}^{2}=0$ in terms of classical 1-absorbing prime submodules. |
| title | On classical 1-absorbing prime submodules |
| topic | Rings and Algebras 13A15, 13C05, 13C13 |
| url | https://arxiv.org/abs/2405.05971 |