Automorphism group of a module

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Definition

Let $R$ be a (unital) ring and $M$ a module over $R$. An automorphism of $M$ (as a $R$-module) is defined as a bijection from $M$ to $M$ that commutes with the action of each element of $R$. The automorphism group of $M$ is defined as the group of all automorphisms of $M$ as a $R$-module. We denote this as $Aut_{R}(M)$.

If $M$ is a free module of order $n$ over $R$, then $Aut_{R}(M)$ can be identified with $GL(n,R)$ (by choosing a freely generating set). In particular, this works when $R$ is a field or a skew field.