Rendiconti del Seminario Matematico della Università di Padova
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Rad-Supplemented ModulesEngin Büyükașik, Engin Mermut and Salahattin Özdemir (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