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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Shahid Bahonar University of Kerman
2025-01-01
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| Series: | Journal of Mahani Mathematical Research |
| Subjects: | |
| Online Access: | https://jmmrc.uk.ac.ir/article_4486_df1c7ba7f361603fb7d51f13823522c4.pdf |
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| Summary: | 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 |
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| ISSN: | 2251-7952 2645-4505 |