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 $R$.