# 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.