Regularity and semipotency of Hom

Abstract

Let $M$, $N$ be modules over a ring $R$ and $[M,N]=\text {Hom}_{R}(M,N)$. The concern is study of: (1) Some fundamental properties of $[M,N]$ when $[M,N]$ is regular or semipotent. (2) The substructures of $[M,N]$ such as radical, the singular and co-singular ideals, the total and others has raised new questions for research in this area. New results obtained include necessary and sufficient conditions for $[M,N]$ to be regular or semipotent. New substructures of $[M,N]$ are studied and its relationship with the $\text {Tot}$ of $[M,N]$. In this paper we show that, the endomorphism ring of a module $M$ is regular if and only if the module $M$ is semi-injective (projective) and the kernel (image) of every endomorphism is a direct summand.