On multiplication fs-modules and dimension symmetry

TL;DR

This paper explores the properties of $fs$-modules, establishing their structure, conditions for being Noetherian, and their behavior over commutative rings, including dimension symmetry and multiplication modules.

Contribution

It introduces new characterizations of $fs$-modules, links them with ring properties, and demonstrates dimension symmetry between modules and their endomorphism rings.

Findings

Every $fs$-module with finite hollow dimension is Noetherian.

An $R$-module with finite Goldie dimension is an $fs$-module iff it decomposes into semisimple and $fs$ parts.

The lattices of submodules coincide for $R$- and $S$-modules, leading to 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 show that $R$ is an $fs$-ring if and only if every multiplication $R$-module is an $fs$-module. In particular, 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 dimension of Krull (Noetherian, Goldie and hollow). 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.