Prime and weakly prime submodules on amalgamated duplication of a ring along an ideal
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| Format: | Preprint |
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2025
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| _version_ | 1866910901775368192 |
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| author | Yeşilot, Gürsel Tarakcı, Esra Şimşek, Yasemin |
| author_facet | Yeşilot, Gürsel Tarakcı, Esra Şimşek, Yasemin |
| contents | Let $A$ be a commutative ring with identity. A proper submodule $N$ of $A$-module $M$ is said to be prime submodule if $ax \in N$ where $a \in A, x \in M$, implies $x \in N$ or $aM \subseteq N$. A proper submodule $N \subset M$ is said to be weakly prime submodule if $0 \neq ax \in N$ where $a \in A, x \in M$, then either $x \in N$ or $aM \subseteq N$. The notion of weakly prime submodule was introduced by Atani and Farzalipour \cite{atani2007weakly}. The purpose of this paper is to study the form of prime and weakly prime submodules of duplication of the $A$-module $M$ along the ideal $I$ (denoted by $M \bowtie I$), introduced and studied by E. M. Bouba, N. Mahdou and M. Tamekkante. A number of results concerning prime and weakly prime submodules on amalgamated duplication and examples are given. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_01633 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Prime and weakly prime submodules on amalgamated duplication of a ring along an ideal Yeşilot, Gürsel Tarakcı, Esra Şimşek, Yasemin Commutative Algebra 13C05 Let $A$ be a commutative ring with identity. A proper submodule $N$ of $A$-module $M$ is said to be prime submodule if $ax \in N$ where $a \in A, x \in M$, implies $x \in N$ or $aM \subseteq N$. A proper submodule $N \subset M$ is said to be weakly prime submodule if $0 \neq ax \in N$ where $a \in A, x \in M$, then either $x \in N$ or $aM \subseteq N$. The notion of weakly prime submodule was introduced by Atani and Farzalipour \cite{atani2007weakly}. The purpose of this paper is to study the form of prime and weakly prime submodules of duplication of the $A$-module $M$ along the ideal $I$ (denoted by $M \bowtie I$), introduced and studied by E. M. Bouba, N. Mahdou and M. Tamekkante. A number of results concerning prime and weakly prime submodules on amalgamated duplication and examples are given. |
| title | Prime and weakly prime submodules on amalgamated duplication of a ring along an ideal |
| topic | Commutative Algebra 13C05 |
| url | https://arxiv.org/abs/2504.01633 |