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On multiplication $fs$-modules and dimension symmetry | ||
| Journal of Mahani Mathematical Research | ||
| دوره 12، شماره 2 - شماره پیاپی 25، مرداد 2023، صفحه 363-374 اصل مقاله (481.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22103/jmmr.2023.20103.1324 | ||
| نویسندگان | ||
| Nasrin Shirali* 1؛ Sayed Malek Javdannezhad2؛ Sayedeh Fatemeh Mousavinasab3 | ||
| 1Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| 2Department of Science, Shahid Rajaee Teacher Training University, Tehran, Iran | ||
| 3Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran | ||
| چکیده | ||
| In this paper, we first study $fs$-modules, i.e., modules with finitely many small submodules. We show that every $fs$-module with finite hollow dimension is Noetherian. Also, we prove that an $R$-module $M$ with finite Goldie dimension, is an $fs$-module if, and only if, $M = M_1 \oplus M_2$, where $M_1$ is semisimple and $M_2$ is an $fs$-module with $Soc(M_2) \ll M$. Then, we investigate multiplication $fs$-modules over commutative rings and we prove that the lattices of $R$-submodules of $M$ and $S$-submodules of $M$ are coincide, where $S=End_R(M)$. Consequently, $M_R$ and $_SM$ have the same Krull (Noetherian, Goldie and hollow) dimension. Further, we prove that for any self-generator multiplication module $M$, to be an $fs$-module as a right $R$-module and as a left $S$-module are equivalent. | ||
| کلیدواژه‌ها | ||
| Small submodules؛ fs-modules؛ multiplication modules؛ dimension symmetry | ||
| مراجع | ||
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