The small condition for modules with Noetherian dimension
A module $M$ with Noetherian dimension is said to satisfy the small condition, if for any small submodule $S$ of $M$ the Noetherian dimension of $S$ is strictly less than the Noetherian dimension of $M$. For an Artinian module $M$, this is equivalent to that $M$ is semisimple. In this article, we i...
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Shahid Bahonar University of Kerman
2025-01-01
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| Series: | Journal of Mahani Mathematical Research |
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| Online Access: | https://jmmrc.uk.ac.ir/article_4486_df1c7ba7f361603fb7d51f13823522c4.pdf |
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| author | Nasrin Shirali Hooriya Kavoosi Ghafi Sayed Malek Javdannezhad |
| author_facet | Nasrin Shirali Hooriya Kavoosi Ghafi Sayed Malek Javdannezhad |
| author_sort | Nasrin Shirali |
| collection | DOAJ |
| description | A module $M$ with Noetherian dimension is said to satisfy the small condition, if for any small submodule $S$ of $M$ the Noetherian dimension of $S$ is strictly less than the Noetherian dimension of $M$. For an Artinian module $M$, this is equivalent to that $M$ is semisimple. In this article, we introduce and study this concept and observe some basic facts for modules with this condition. As a main result, it is shown that if $M$ is a module with finite hollow dimension which satisfies the small condition, then $\alpha \leq n-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ n-dim\,S: S\ll M\}$. Furthermore, if $M$ is a module with Krull dimension and finite hollow dimension, then $\alpha \leq k-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ k-dim\,S: S\ll M\}$. Also, we study the projective cover of modules satisfying the small condition or with finite hollow dimension |
| format | Article |
| id | doaj-art-fe7af37aaa434c0cbc33da645c723039 |
| institution | Kabale University |
| issn | 2251-7952 2645-4505 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Shahid Bahonar University of Kerman |
| record_format | Article |
| series | Journal of Mahani Mathematical Research |
| spelling | doaj-art-fe7af37aaa434c0cbc33da645c7230392025-01-04T19:30:18ZengShahid Bahonar University of KermanJournal of Mahani Mathematical Research2251-79522645-45052025-01-0114132734310.22103/jmmr.2024.23633.16704486The small condition for modules with Noetherian dimensionNasrin Shirali0Hooriya Kavoosi Ghafi1Sayed Malek Javdannezhad2Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, IranDepartment of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, IranDepartment of Mathematics, Shahid Rajaee Teacher Training University, Tehran, IranA module $M$ with Noetherian dimension is said to satisfy the small condition, if for any small submodule $S$ of $M$ the Noetherian dimension of $S$ is strictly less than the Noetherian dimension of $M$. For an Artinian module $M$, this is equivalent to that $M$ is semisimple. In this article, we introduce and study this concept and observe some basic facts for modules with this condition. As a main result, it is shown that if $M$ is a module with finite hollow dimension which satisfies the small condition, then $\alpha \leq n-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ n-dim\,S: S\ll M\}$. Furthermore, if $M$ is a module with Krull dimension and finite hollow dimension, then $\alpha \leq k-dim\, M\leq \alpha+1$, where $\alpha=\sup\{ k-dim\,S: S\ll M\}$. Also, we study the projective cover of modules satisfying the small condition or with finite hollow dimensionhttps://jmmrc.uk.ac.ir/article_4486_df1c7ba7f361603fb7d51f13823522c4.pdflarge conditionsmall conditionsemiatomic modulesprojective cover |
| spellingShingle | Nasrin Shirali Hooriya Kavoosi Ghafi Sayed Malek Javdannezhad The small condition for modules with Noetherian dimension Journal of Mahani Mathematical Research large condition small condition semiatomic modules projective cover |
| title | The small condition for modules with Noetherian dimension |
| title_full | The small condition for modules with Noetherian dimension |
| title_fullStr | The small condition for modules with Noetherian dimension |
| title_full_unstemmed | The small condition for modules with Noetherian dimension |
| title_short | The small condition for modules with Noetherian dimension |
| title_sort | small condition for modules with noetherian dimension |
| topic | large condition small condition semiatomic modules projective cover |
| url | https://jmmrc.uk.ac.ir/article_4486_df1c7ba7f361603fb7d51f13823522c4.pdf |
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