Find an isomorphism $D: Hom_R(R^p, R^q) \longrightarrow R^{p\times q}$

Question

I know this would be really basic, but I'm currently struggling to describe the proper isomorphism that fulfills the following description for natural $p$ and $q$: $$D: Hom_R(R^p, R^q) \longrightarrow R^{p\times q}$$

I can't find one, plus how would one prove it's an isomorphism (I'm more sure here: we just need to prove it's a linear function and a bijective one). Many thanks in advance!

Answer 1

Let $e_i$ for $i<p$ be a base for $R^p$ and $\pi_j:R^q\to R$ for $j<q$ be the canonical projection. Then \begin{align} &\operatorname{Hom}(R^p,R^q)\to R^{p\times q}& &\varphi\mapsto((\pi_j\circ\varphi)(e_i):i<p\land j<q) \end{align} is a module isomorphism.