Rad-Supplemented Modules

Abstract

Let $\tau$ be a radical for the category of left $R$-modules for a ring $R$. If $M$ is a $\tau$-coatomic module, that is, if $M$ has no nonzero $\tau$-torsion factor module, then $\tau (M)$ is small in $M$. If $V$ is a $\tau$-supplement in $M$, then the intersection of $V$ and $\tau (M)$ is $\tau (V)$. In particular, if $V$ is a $\mathrm{Rad}$-supplement in $M$, then the intersection of $V$ and $\mathrm{Rad}(M)$ is $\mathrm{Rad}(V)$. A module $M$ is $\tau$-supplemented if and only if the factor module of $M$ by $P_{\tau }(M)$ is $\tau$-supplemented where $P_{\tau }(M)$ is the sum of all $\tau$-torsion submodules of $M$. Every left $R$-module is $\mathrm{Rad}$-supplemented if and only if the direct sum of countably many copies of $R$ is a $\mathrm{Rad}$-supplemented left $R$-module if and only if every reduced left $R$-module is supplemented if and only if $R/P(R)$ is left perfect where $R/P(R)$ is the sum of all left ideals $I$ of $R$ such that $\mathrm{Rad}I=I$. For a left duo ring $R$, $R$ is a $\mathrm{Rad}$-supplemented left $R$-module if and only if $R/P(R)$ is semiperfect. For a Dedekind domain $R$, an $R$-module $M$ is $\mathrm{Rad}$-supplemented if and only if $M/D$ is supplemented where $D$ is the divisible part of $M$.