Generalization of the theorems of Johnson and Utumi to \(\text{Hom}\) (Q1914716)

Let \(R\) be an associative ring with identity, \(M\) and \(N\) unital left \(R\)-modules, \(S=\text{End}_R(N)\) and \(T=\text{End}_R(M)\). The main purpose of this paper is to extend the well-known theorems of Johnson and Utumi concerning the endomorphism rings of self-injective modules and self-projective modules, from End to Hom. The main ingredient in this approach is the concept of Kelly-radical \(\text{RAD}(M,N)\) of \(M\) and \(N\), which is the set set of all \(f\in\text{Hom}_R(M,N)\) such that for all \(g\in\text{Hom}_R(N,M)\) the morphism \(1_M-gf\) is invertible in \(T\). Clearly, when \(M=N\), then \(\text{RAD}(M,N)\) is precisely the Jacobson radical of the ring \(T\). For instance, it is proved that if \(M\) is self-injective and \(N\)-injective, then for any \(f\in\text{Hom}_R(M,N)\) there exists a \(g\in\text{Hom}_R(N,M)\) such that \(fgf-f\in\text{RAD}(M,N)\). As applications, the author investigates when \(\text{RAD}(M,N)\) equals the total \(\text{TOT}(M,N)\) on one hand, and the Jacobson radical of the modules \(_S\text{Hom}_R(M,N)\) and respectively \(\text{Hom}_R(M,N)_T\) on the other hand. In general, \(\text{RAD}(M,N)\) is a subset of \(\text{TOT}(M,N)\), which is the set of all \(f\in\text{Hom}_R(M,N)\) such that for all \(g\in\text{Hom}_R(N,M)\), \((gf)^2=gf\) implies \(gf=0\). For instance, if \(M\) is self-injective and \(N\)-injective, then \(\text{RAD}(M,N)=\text{TOT}(M,N)\).