I -Semipotency and the Total of Modules

The object of this paper is to study the total as substructure of hom (M,N) Rfor any two modules R M and R N , one of interesting question, is when the totalof a module N equals the hom (N, J (N)) R . Toward this question, manyresults have been obtained, where we characterize the module N and for whichTot[M,N] = I [M,N] for all M Îmod - R . The main obtained resultsinclude:(a) Let R N be module. The following conditions are equivalent:(1) The module R N is an I - module.(2) Tot[M,N] = I [M,N] for all M Îmod - R .(3) Tot[N,M] = I [N,M] for all M Îmod - R .(4) [N,M] is a I - semipotent for all M Îmod - R .(5) [M,N] is a I - semipotent for all M Îmod - R .(6) N E is an I - semipotent ring and for all M Îmod - R ,I [M, N] { : [M,N]; I (E ) for any [N, M] } N = a a Î ab Î b Î(b) Let R N be an I - module with G(N) = {0} . The following conditionsare equivalent:(1) N E is a semipotent ring.(2) ( ) ( ) N N J E = I E .(3) For every ( ) N N a Î E J E there exists a projective direct summand0 ¹W ÍÅ N such that W Í Im(a)