Invertible Matrix corresponds to Automorphism

Theorem
Let $$R$$ be a ring with unity.

Let $$G$$ be an $n$-dimensional $R$-module.

Let $$\mathcal {M}_{R} \left({n}\right)$$ be the $n \times n$ matrix space over $$R$$.

Let $$\mathcal {L}_R \left({G}\right)$$ be the set of all linear operators on $$G$$.

Then the invertible elements of the ring $\left({\mathcal {M}_{R} \left({n}\right), +, \times}\right)$ correspond directly to automorphisms of $$\mathcal {L}_R \left({G}\right)$$.