On multiplication $fs$-modules and dimension symmetry

Document Type : Research Paper

Authors

1 Department of Mathematics, Shahid Chamran University of Ahvaz, Ahvaz, Iran

2 Department of Science, Shahid Rajaee Teacher Training University, Tehran, Iran

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.

Keywords

References

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History

  • Receive Date: 20 August 2022
  • Revise Date: 25 December 2022
  • Accept Date: 19 January 2023

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