Let M and N be right R-modules. Hom(M, N) is called regular if for each fHom(M, N), there exists gHom(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 MN is locally split. In this article, we also study the substructures of Hom(M, N) such as the Jacobson radical J[M, N], the singular ideal [M, N], and the co-singular ideal delta[M, N]. We prove several new results. The question is to characterize when the Jacobson radical is equal to the singular ideal [M, N] or the co-singular ideal delta[M, N] under injectivity and projectivity.