Invertible Matrix corresponds to Automorphism

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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 $\mathcal L_R \left({G}\right)$.


Proof


Sources