Endomorphisms

A module endomorphism is a module homomorphism from a module to itself.

Definition: Endomorphism Ring

If $M$ is an $R$-module, then we define the endomorphism ring of $M$ as ${\text{End}}_R(M)=\{R\text{-module homomorphisms }M\to M\}.$ This is a subring of the homomorphism ring ${\text{Hom}}_{\text{Ab-Gp}}(M,M)$.