On graded weakly $ J_{gr} $-semiprime submodules
Let $ \Gamma $ be a group, $ \mathcal{A} $ be a $ \Gamma $-graded commutative ring with unity $ 1, $ and $ \mathcal{D} $ a graded $ \mathcal{A} $-module. In this paper, we introduce the concept of graded weakly $ J_{gr} $-semiprime submodules as a generalization of graded weakly semiprime submodules...
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| Language: | English |
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AIMS Press
2024-03-01
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| Series: | AIMS Mathematics |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/math.2024602?viewType=HTML |
| _version_ | 1797206052356227072 |
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| author | Malak Alnimer Khaldoun Al-Zoubi Mohammed Al-Dolat |
| author_facet | Malak Alnimer Khaldoun Al-Zoubi Mohammed Al-Dolat |
| author_sort | Malak Alnimer |
| collection | DOAJ |
| description | Let $ \Gamma $ be a group, $ \mathcal{A} $ be a $ \Gamma $-graded commutative ring with unity $ 1, $ and $ \mathcal{D} $ a graded $ \mathcal{A} $-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 $ \mathcal{D} $ is said to be a graded weakly $ J_{gr} $-semiprime submodule of $ \mathcal{D} $ if whenever $ r_{g}\in h(\mathcal{A}), $ $ m_{h}\in h(\mathcal{D}) $ and $ n\in \mathbb{Z} ^{+} $ with $ 0\neq r_{g}^{n}m_{h}\in U $, then $ r_{g}m_{h}\in U+J_{gr}(\mathcal{D}) $, where $ J_{gr}(\mathcal{D}) $ is the graded Jacobson radical of $ \mathcal{D}. $ |
| first_indexed | 2024-04-24T09:00:53Z |
| format | Article |
| id | doaj.art-86b192e5de3a48fa8c7d97b842328e2c |
| institution | Directory Open Access Journal |
| issn | 2473-6988 |
| language | English |
| last_indexed | 2024-04-24T09:00:53Z |
| publishDate | 2024-03-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | AIMS Mathematics |
| spelling | doaj.art-86b192e5de3a48fa8c7d97b842328e2c2024-04-16T01:15:12ZengAIMS PressAIMS Mathematics2473-69882024-03-0195123151232210.3934/math.2024602On graded weakly $ J_{gr} $-semiprime submodulesMalak Alnimer 0Khaldoun Al-Zoubi1Mohammed Al-Dolat2Department of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanDepartment of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanDepartment of Mathematics and Statistics, Jordan University of Science and Technology, P.O. Box 3030, Irbid 22110, JordanLet $ \Gamma $ be a group, $ \mathcal{A} $ be a $ \Gamma $-graded commutative ring with unity $ 1, $ and $ \mathcal{D} $ a graded $ \mathcal{A} $-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 $ \mathcal{D} $ is said to be a graded weakly $ J_{gr} $-semiprime submodule of $ \mathcal{D} $ if whenever $ r_{g}\in h(\mathcal{A}), $ $ m_{h}\in h(\mathcal{D}) $ and $ n\in \mathbb{Z} ^{+} $ with $ 0\neq r_{g}^{n}m_{h}\in U $, then $ r_{g}m_{h}\in U+J_{gr}(\mathcal{D}) $, where $ J_{gr}(\mathcal{D}) $ is the graded Jacobson radical of $ \mathcal{D}. $https://www.aimspress.com/article/doi/10.3934/math.2024602?viewType=HTMLgraded weakly $ j_{gr} $-semiprime submodulegraded $ j_{gr} $-semiprime submodulegraded weakly semiprime submodule |
| spellingShingle | Malak Alnimer Khaldoun Al-Zoubi Mohammed Al-Dolat On graded weakly $ J_{gr} $-semiprime submodules AIMS Mathematics graded weakly $ j_{gr} $-semiprime submodule graded $ j_{gr} $-semiprime submodule graded weakly semiprime submodule |
| title | On graded weakly $ J_{gr} $-semiprime submodules |
| title_full | On graded weakly $ J_{gr} $-semiprime submodules |
| title_fullStr | On graded weakly $ J_{gr} $-semiprime submodules |
| title_full_unstemmed | On graded weakly $ J_{gr} $-semiprime submodules |
| title_short | On graded weakly $ J_{gr} $-semiprime submodules |
| title_sort | on graded weakly j gr semiprime submodules |
| topic | graded weakly $ j_{gr} $-semiprime submodule graded $ j_{gr} $-semiprime submodule graded weakly semiprime submodule |
| url | https://www.aimspress.com/article/doi/10.3934/math.2024602?viewType=HTML |
| work_keys_str_mv | AT malakalnimer ongradedweaklyjgrsemiprimesubmodules AT khaldounalzoubi ongradedweaklyjgrsemiprimesubmodules AT mohammedaldolat ongradedweaklyjgrsemiprimesubmodules |