On multiplication $fs$-modules and dimension symmetry

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Shirali Nasrin; Javdannezhad Sayed Malek; Mousavinasab Sayedeh Fatemeh

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Journal Paper

Paper Information

Journal: Journal of Mahani Mathematical Research Year:2023 | Volume:12 | Issue:2 Page(s): 363-374

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Title

On multiplication $fs$-modules and dimension symmetry

Author(s)

Shirali Nasrin | Javdannezhad Sayed Malek | Mousavinasab Sayedeh Fatemeh | Issue Writer Certificate

Keywords

Small submodules

fs-modules

multiplication modules

dimension symmetry

Abstract

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.

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