ON F-SUPPLEMENTED MODULES

Abstract

Let $R$ be a ring and $M$ a right $R$–module. In this paper we prove that if $M$ is weakly $\mathrm{F}$-supplemented, then every factor module and every $\mathrm{F}$-coclosed submodule of $M$ is again weakly $\mathrm{F}$-supplemented. In [5], it is shown that $Rad\left(M\right)$ has finite uniform dimension iff $M$ does not contain an infinite direct sum of nonzero small submodules. Here we replace $\mathrm{F}$-small submodules instead of small submodules (which is a weaker condition) and obtain the same result; i.e, we show that if $M$ does not contain an infinite direct sum of $\mathrm{F}$-small submodules, then $Rad\left(M\right)$ has finite uniform dimension.