General linear group over a commutative unital ring

Definition

This is a generalization to commutative unital rings of the notion of general linear group over a field.

In terms of dimensions (finite case)

Let $R$ be a commutative unital ring and $n$ a natural number. The general linear group of degree $n$ over $R$, denoted $GL(n,R)$ or $GL_{n}(R)$is defined in the following equivalent ways:

• $GL(n,R)$ is the group of $R$-module automorphisms from the free $R$-module $R^{n}$ to itself.
• $GL(n,R)$ is the group of invertible $n\times n$ matrices with entries in $R$, under matrix multiplication.

In terms of free modules

Let $R$ be a commutative unital ring and $M$ be a free module over $R$. The general linear group $GL(M)$ is defined as the group of automorphisms of $M$ as a $R$-module.

More general versions

• Automorphism group of a projective module over a commutative unital ring: General linear groups are the automorphism groups of (finitely generated in the matrix case) free modules. We may be interested in similarly studying the automorphism groups of finitely generated projective modules.
• General linear group over a unital ring