Good Conditions for the Total
Let R be a ring with 1 ∈ R. In Mod-R the total Tot(M, N) is a semi-ideal, which contains the radical Rad(M, N), the singular ideal △(M,N)and the cosingular ideal ▽(M, N). We study conditions on modules Q and P, which imply that Rad(Q, N) = △(Q,N) = Tot(Q, N) and Rad(M, P) = ▽(M, P) = Tot(M, P) for all M and N. We prove that these equalities hold if Q is injective, resp. P is semi-perfect and projective. To get further results and interesting topics, we consider the question: For which Q is △(Q,N) = Tot(Q, N) for all N? Here we study rings R such that the condition △(Q, N) = Tot(Q, N) for all N implies that Q is a direct sum of injective modules. We conjecture that such rings must be right Noetherian and prove that they (and all their homomorphic images) are right Goldie rings. Further, the conjecture is confirmed in a number of cases.
KeywordsDirect Summand Injective Module Noetherian Ring Primitive Ideal Minimal Prime Ideal
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