General Linear Group - in Terms of Determinants

In Terms of Determinants

Over a field F, a matrix is invertible if and only if its determinant is nonzero. Therefore an alternative definition of GL(n, F) is as the group of matrices with nonzero determinant.

Over a commutative ring R, one must be slightly more careful: a matrix over R is invertible if and only if its determinant is a unit in R, that is, if its determinant is invertible in R. Therefore GL(n, R) may be defined as the group of matrices whose determinants are units.

Over a non-commutative ring R, determinants are not at all well behaved. In this case, GL(n, R) may be defined as the unit group of the matrix ring M(n, R).