# Rad-Supplemented Modules

### Engin Büyükașik

Izmir Institute of Technology, Turkey### Engin Mermut

Dokuz Eylül Üniversiteși, Buca/izmir, Turkey### Salahattin Özdemir

Dokuz Eylül Üniversiteși, Buca/izmir, Turkey

## Abstract

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

## Cite this article

Engin Büyükașik, Engin Mermut, Salahattin Özdemir, Rad-Supplemented Modules. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 157–177

DOI 10.4171/RSMUP/124-10