Semipotency and the total of rings and modules

Let M and N be two modules over a ring R. The object of this paper is the study of substructures of HomR(M,N) such as, radical, the singular, and co-singular ideal and the total. New results obtained include necessary and sufficient conditions for the total to equal the radical, HomR(M,J(N)), a description of (Δ-, ∇-, I-) semipotency rings and the endomorphism ring of locally projective module. New structure theorems are obtained by studying the relationship between two concepts of the total and (Δ-, ∇-, I-) semi-potentness. In addition, locally injective and locally projective modules are characterized in new ways.