# Definition:Module of Homomorphisms Between Modules

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## Definition

Let $R$ be a commutative ring.

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

Let $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ denote the set of $R$-module homomorphisms from $M$ to $N$.

Let:

$+$ be the operation on $\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}$ defined by $f + g: m \mapsto \map f m + \map g m$
$\circ$ be defined as $\lambda \circ f: m \mapsto \lambda \map f m$

Then $\struct {\map {\operatorname {Hom}_{R - \operatorname {mod} } } {M, N}, + , \circ}$ is called the module of homomorphisms between $M$ and $N$.