Editing General linear group (section)

== In terms of determinants ==

Over a field <math>F</math>, a matrix is [[invertible]] if and only if its [[determinant]] is nonzero. Therefore, an alternative definition of <math>\operatorname{GL}(n,F)</math> is as the group of matrices with nonzero determinant.

Over a [[commutative ring]] <math>R</math>, more care is needed: a matrix over <math>R</math> is invertible if and only if its determinant is a [[unit (ring theory)|unit]] in <math>R</math>, that is, if its determinant is invertible in <math>R</math>. Therefore, <math>\operatorname{GL}(n,R)</math> may be defined as the group of matrices whose determinants are units.

Over a non-commutative ring <math>R</math>, determinants are not at all well behaved. In this case, <math>\operatorname{GL}(n,R)</math> may be defined as the [[unit group]] of the [[matrix ring]] <math>M(n,R)</math>.