Homomorphism Rings

Let $M$ be an abelian group. The set ${\text{Hom}}_{\text{Ab-Gp}}(M,M)=\{\text{group homomorphisms }M\to M\}$ is a ring where

• $\varphi +\psi$ is defined by $(\varphi +\psi )(m)=\varphi (m)+\psi (m)$
• $\varphi \psi$ is defined by $\varphi \psi (m)=\varphi \circ \psi (m)$ (i.e., composition)

A natural bijection exists between $\{\text{left R-module structures on M}\}⟷\{\text{ring homomorphisms }\rho :R\to {\text{Hom}}_{\text{Ab-Gp}}(M,M)\}$