On graded weakly $J_{gr}$-semiprime submodules
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866917624939544576 |
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| author | Alnimer, Malak Al-Zoubi, Khaldoun Al-Dolat, Mohammed |
| author_facet | Alnimer, Malak Al-Zoubi, Khaldoun Al-Dolat, Mohammed |
| contents | Let $Γ$ be a group, $\Re$ be a $Γ$-graded commutative ring with unity $1$ and $\Im$ a graded $\Re$-module. In this paper, we introduce the concept of graded weakly $J_{gr}$-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $J_{gr}$% -semiprime submodules. For example, we give a characterization of graded weakly $J_{gr}$-semiprime submodules. Also, we find some relations between graded weakly $J_{gr}$-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules are investigated. A proper graded submodule $U$ of $\Im$ is said to be a graded weakly $J_{gr}$-semiprime submodule of $\Im$ if whenever $r_{g}\in h(\Re),$ $m_{h}\in h(\Im)$ and $n\in %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$, then $r_{g}m_{h}\in U+J_{gr}(\Im)$, where $J_{gr}(\Im)$ is the graded Jacobson radical of $\Im.$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_08858 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On graded weakly $J_{gr}$-semiprime submodules Alnimer, Malak Al-Zoubi, Khaldoun Al-Dolat, Mohammed General Mathematics Let $Γ$ be a group, $\Re$ be a $Γ$-graded commutative ring with unity $1$ and $\Im$ a graded $\Re$-module. In this paper, we introduce the concept of graded weakly $J_{gr}$-semiprime submodules as a generalization of graded weakly semiprime submodules. We study several results concerning of graded weakly $J_{gr}$% -semiprime submodules. For example, we give a characterization of graded weakly $J_{gr}$-semiprime submodules. Also, we find some relations between graded weakly $J_{gr}$-semiprime submodules and graded weakly semiprime submodules. In addition, the necessary and sufficient condition for graded submodules to be graded weakly $J_{gr}$-semiprime submodules are investigated. A proper graded submodule $U$ of $\Im$ is said to be a graded weakly $J_{gr}$-semiprime submodule of $\Im$ if whenever $r_{g}\in h(\Re),$ $m_{h}\in h(\Im)$ and $n\in %TCIMACRO{\U{2124} }% %BeginExpansion \mathbb{Z} %EndExpansion ^{+}$ with $0\neq r_{g}^{n}m_{h}\in U$, then $r_{g}m_{h}\in U+J_{gr}(\Im)$, where $J_{gr}(\Im)$ is the graded Jacobson radical of $\Im.$ |
| title | On graded weakly $J_{gr}$-semiprime submodules |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2305.08858 |