Matrix Space is Module

Theorem
Let $$\left({R, +, \circ}\right)$$ be a ring.

Let $$\mathbf{A} = \left[{a}\right]_{m n}$$ be an $m \times n$ matrix over $$\left({R, +, \circ}\right)$$.

Then the matrix space $\mathcal {M}_{R} \left({m, n}\right)$ of all $m \times n$ matrices over $$R$$ is a module.

Proof
This follows as $$\mathcal {M}_{R} \left({m, n}\right)$$ is a direct instance of the module given in the Module of All Mappings, where $$\mathcal {M}_{R} \left({m, n}\right)$$ is the $$R$$-module $$R^{\left[{1 \,. \, . \, m}\right] \times \left[{1 \,. \, . \, n}\right]}$$.

The $$S$$ of that example is the set $$\left[{1 \,. \, . \, m}\right] \times \left[{1 \,. \, . \, n}\right]$$, while the $$G$$ of that example is the $$R$$-module $$R$$.