Let $R$ be a commutative ring, and let $M$ and $N$ be left $R$-modules. Let $\operatorname{Hom}_R(M,N)$ denote the set of $R$-module homomorphisms $M\to N$. Define addition and scalar multiplication on $\operatorname{Hom}_R(M,N)$ by
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\begin{align*}
(f+g)(m)=f(m)+g(m)
\end{align*}
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for all $f,g\in \operatorname{Hom}_R(M,N)$ and $m\in M$, and
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\begin{align*}
(rf)(m)=r f(m)
\end{align*}
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for all $r\in R$, $f\in \operatorname{Hom}_R(M,N)$, and $m\in M$. Then $\operatorname{Hom}_R(M,N)$ is a left $R$-module.