Substructures Of [M,N]

Let RM , RN be two modules over a ring R and ),(],[ NMomhNM R= , then ],[ NM is an −),( MN EE bimodule. The concern is about the five substructures of ),( NMomh R : the Jacobson radical ],[ NMJ , the singular ideal ],[ NM∆ , the co-singular ideal ],[ NM∇ , the total ],[Tot NM and the ],[ NMI . One natural question is to characterize when the total is equal to one or more of the other structures. Toward this question, many results have been obtained: (a) A module RW is locally projective if and only if, ],[],[Tot WMWM ∇= for every quasi-projective module RM −∈ mod . (b) A module RQ is locally injective if and only if, ],[],[Tot NQNQ ∆= for every quasi-injective module RN −∈ mod . (c) A module RN with }0{)( =Γ N is an −I module if and only if, ],[],[Tot NMINM = for every module RM −∈ mod . (d) A module RN with }0{)( =Γ N is an −I module if and only if, for every NE∈α with )()1( NJmI ⊆−α , is one-to-one