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The small condition for modules with Noetherian dimension | ||
| Journal of Mahani Mathematical Research | ||
| دوره 14، شماره 1 - شماره پیاپی 31، فروردین 2025، صفحه 327-343 اصل مقاله (537.32 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2024.23633.1670 | ||
| نویسندگان | ||
| Nasrin Shirali* 1؛ Hooriya Kavoosi Ghafi1؛ Sayed Malek Javdannezhad2 | ||
| 1Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| 2Department of Mathematics, Shahid Rajaee Teacher Training University, Tehran, Iran | ||
| چکیده | ||
| 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 | ||
| کلیدواژه‌ها | ||
| Large condition؛ Small condition؛ Semiatomic modules؛ Projective cover | ||
| مراجع | ||
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