# Defining a right $A$-module structure on $\text{Hom}_\mathbb{Z}(E,G)$

Let $$A$$ be a ring. Suppose $$E$$ is a right $$A$$-module, $$F$$ a left $$A$$-module and $$G$$ a $$\mathbb{Z}$$-module. Is there a way to define a right $$A$$-module structure on $$\text{Hom}_\mathbb{Z}(E,G)$$?

This would equivalent to defining a ring homomorphism $$\phi:A^{\text{op}}\rightarrow\mathcal{E}$$, where $$\mathcal{E}$$ is the endomorphism ring of $$\text{Hom}_\mathbb{Z}(E,G)$$. For $$a\in A$$, what should $$\phi_a:\text{Hom}_\mathbb{Z}(E,G)\rightarrow \text{Hom}_\mathbb{Z}(E,G)$$ be?

You define $$F$$ but it doesn't appear in your question so I'm not sure you've asked the question you meant to ask. The general pattern is that $$\text{Hom}$$ is a contravariant functor in its first argument so it switches left and right module structures: $$\text{Hom}_{\mathbb{Z}}(E, A)$$ (for $$A$$ an abelian group; IMO it's good practice to reserve $$G$$ for a not-necessarily-abelian group) naturally acquires a left $$A$$-module structure by precomposition, and similarly $$\text{Hom}_{\mathbb{Z}}(F, A)$$ naturally acquires a right $$A$$-module structure by precomposition.