[[Module theory MOC]]
# Hom-module

Let $R$ be a [[commutative ring]] and $U,V$ be $R$-[[module|modules]].
Then the set $\lMod{R}(U,V)$ of $R$-[[module homomorphism|module homomorphisms]] is itself equipped with the structure of an $R$-module, i.e. we have an [[Closed category|internal hom-functor]]
$$
\begin{align*}
\lMod R(-,-) : \op{\lMod R} \times \lMod R \to \lMod R
\end{align*}
$$
which is the [[Adjoint functor|right adjoint]] of the [[Tensor product of modules over a commutative ring|tensor product]], i.e.
$$
\begin{align*}
(-) \otimes_{R} V \dashv \lMod R(V,-)
\end{align*}
$$

> [!missing]- Proof
> #missing/proof

#
---
#state/develop | #lang/en | #SemBr