Let be a radical for the category of left R-modules for a ring R. If M is a -coatomic module, that is, if M has no nonzero -torsion factor module, then (M) is small in M. If V is a -supplement in M, then the intersection of V and τ(M) is (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 -supplemented if and only if the factor module of M by P(M) is -supplemented where P(M) is the sum of all -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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Engin Büyükașik, Engin Mermut, Salahattin Özdemir, Rad-Supplemented Modules. Rend. Sem. Mat. Univ. Padova 124 (2010), pp. 157–177DOI 10.4171/RSMUP/124-10