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).
Read more about this topic: General Linear Group
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