General linear group over a commutative unital ring

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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