# Definition:Homomorphism Ring between Modules

## Definition

Let $R$ be a ring.

Let $M,N$ be $R$-modules.

The set of all $R$-module homomorphisms from $M$ to $N$ forms a ring, denoted

- $\operatorname{Hom}_R(M,N)$

or just

- $\operatorname{Hom}(M,N)$

if the ring $R$ is clear from the context.