on multiplication fs-modules and dimension symmetry

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.