On (I-) Semipotent [M,N]

Let MR and NR be modules, we use [M,N] = hom (M,N), so [M,N] is an (EM, EN) - bimodule. Some of the interesting questions are when the total equals the Jacobson radical, ▲ EM, VEM and I(Em) for some module M, In this paper we study the question is when the total equals the ideal I(EM). New results obtained include: (1) A projective module PR is an I-module if and only if E, = End (P) is an I - semipotent. (2) A ring • R P R R is a semipotent ring if and only if, endomorphism ring of any projective module PR is an I - semipotent. (3) For any projective module PR; Tot (Ep)= I(Ep) if and only if, P is an I。 module. (4) For any ring R; Tot (R) = J(R) if and only if, [M,P] is an I – semipotent for any projective module PR and any module MR which equivalent that, Tot [M,P] = I[M,P] for any projective module P and any module M which also, equivalent [M,P] is semipotent for any finitely generated projetive module PR and any module M