General linear group over a commutative unital ring
In terms of dimensions (finite case)
Let be a commutative unital ring and a natural number. The general linear group of degree over , denoted or is defined in the following equivalent ways:
- is the group of -module automorphisms from the free -module to itself.
- is the group of invertible matrices with entries in , under matrix multiplication.
In terms of free modules
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