On (Semi)Regular Morphisms

Let M and N be right R-modules. Hom(M, N) is called regular if for each f ∈ Hom(M, N), there exists g ∈ Hom(N, M) such that f = fgf. Let M, N] = Hom _R (M, N). We prove that if M is finitely generated, then M, N] is regular if and only if every homomorphism M → N is locally split. In this article, we also study the substructures of Hom(M, N) such as the Jacobson radical JM, N], the singular ideal ΔM, N], and the co-singular ideal ?M, N]. We prove several new results. The question is to characterize when the Jacobson radical is equal to the singular ideal ΔM, N] or the co-singular ideal ?M, N] under injectivity and projectivity.