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 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.