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

Let:

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


Then $\left(\operatorname{Hom}_{R-\text{mod}}(M,N),+,\circ\right)$ is called the module of homomorphisms between $M$ and $N$.


Also see