SUPPLEMENTED MORPHISMS

In the present paper, left R-modules M and N are studied under the assumptions that Hom_R(M,N) is supplemented. It is shown that Hom(M,N) is (⊕, G*, amply)-supplemented if and only if N is (⊕, G*, amply)-supplemented. Some applications to cosemisimple modules, refinable modules and UCC-modules are presented. Finally, the relationship between the Jacobson radical J[M,N] of Hom_R(M,N) and Hom_R(M,N) is supplemented are investigated. Let M be a finitely generated, self-projective left R-module and N ∈ Gen(M). We show that if Hom(M,N) is supplemented and N has GD2 then Hom(M,N)/J(M,N) is semisimple as a left E_M-module.