# Rendiconti del Seminario Matematico della Università di Padova

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**Volume 124, 2010, pp. 157–177**

**DOI: 10.4171/RSMUP/124-10**

Rad-Supplemented Modules

Engin Büyükașik^{[1]}, Engin Mermut

^{[2]}and Salahattin Özdemir

^{[3]}(1) Department of Mathematics, Izmir Institute of Technology, Urla, 35430, Izmir, Turkey

(2) Department of Mathematics, Dokuz Eylül Üniversiteși, Tinaztepe Yerleșkesi, 35160, Buca/izmir, Turkey

(3) Department of Mathematics, Dokuz Eylül Üniversiteși, Tinaztepe Yerleșkesi, 35160, Buca/izmir, Turkey

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 τ(*M*) is $\tau$(*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 $\tau$-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 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 Rad *I* = *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*.

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Büyükașik Engin, Mermut Engin, Özdemir Salahattin: Rad-Supplemented Modules. *Rend. Sem. Mat. Univ. Padova* 124 (2010), 157-177. doi: 10.4171/RSMUP/124-10