SUBSTRUCTURES OF HOM.

Let M and N be two modules over a ring R. The concern is about the four substructures of homR(M, N): the Jacobson radical J[M, N], the singular ideal Δ[M, N], the co-singular ideal ∇[M, N] and the total Tot[M, N]. One natural question is to characterize when the total is equal to one or more of the other structures. We review some known results and prove several new results towards this question and, as consequences, give answers to some related questions. [ABSTRACT FROM AUTHOR]