forgetful functor from RMod to Ab as right adjoint
1. Proposition
Let \(R\) be a ring and \(\mathrm{RMod}\) the category RMod and \(\mathrm{Ab}\) the Category Ab. Then the forgetful functor from RMod to Ab is a right adjoint to the tensoring to change the ring as functor
2. Proof
Let \(A \in \mathrm{Ob}(\mathrm{Ab})\) be a an abelian group and \(M\) an \(R\)-module. Then we construct
\begin{align*} \varphi: \mathrm{Hom}_{\mathrm{RMod}}(R \otimes_{\mathbb{Z}} A, M) \rightarrow& \mathrm{Hom}_{\mathrm{Ab}}(A,M) \\ (f: R \otimes_{\mathbb{Z}} A \rightarrow M) \mapsto& (f': a \mapsto f(1 \otimes_{\mathbb{Z}} a)) \\ \psi: \mathrm{Hom}_{\mathrm{Ab}}(A,M) \rightarrow& \mathrm{Hom}_{\mathrm{RMod}}(R \otimes_{\mathbb{Z}} A, M) \\ (f: A \rightarrow M) \mapsto& f: ((r \otimes_{\mathbb{Z}} a) \mapsto r \cdot f(a)) \end{align*}