covariant hom functor from Ab to RMod

Let \(R\) be a ring. Then the covariant hom functor is defined as functor

\begin{align*} \mathrm{Hom}_{\mathrm{Ab}}(R,-): \mathrm{Ab} \rightarrow& \mathrm{RMod} \\ M \mapsto& \mathrm{Hom}_{\mathrm{Ab}}(R,M) \\ \end{align*}

see:

• covariant hom functor from Grp to RMod
• where \(\mathrm{Hom}_{\mathrm{Ab}}(R,A) = \mathrm{Hom}_{\mathrm{Grp}}(R,A)\) for \(A \in \mathrm{Ob}(\mathrm{Ab})\), as Category Ab is a full subcategory