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The Hopf envelope of the K-bialgebra associated to the multiplication of
invertible n x n - matrices over associative algebras is given as residue class
algebra of the free algebra  K< X^(h)_ij : i,j=1,...,n, h = 0,1,2,... >
in non-commuting variables X^(h)_ij modulo the ideal I generated by
\sum_{k=1}^n X^(h)_ik X^(h+1)_jk - \delta_ij,
\sum_{k=1}^n X^(h+1)_ki X^(h)_kj - \delta_ij, all i,j,h.
It is representing a functor Q=QGL_n, which is a subfunctor
(called quantum general linear group) of
the functor general linear group of n x n - matrices
GL_n: (associative algebras with 1) -> (sets).

We also study canonical subfunctors Q*=Q*GL_n, Q^m of GL_n and give
(noncommutative) Gröbner bases for the ideals in K< X^(h)_ij : i,j=1,...,n, h = 0,1,2,... >
which define the coordinate algebras O(Q), O(Q*), O(GL_n).
O(Q*) is the envelope of O(GL_n) in the class of Hopf algebras with bijective
antipode. O(Q^m) is the general residue class Hopf algebra of O(Q*) whose
antipode has order 2m. Our main result is a degree formula for A=
O(Q), O(Q*), O(GL_n). It follows that there are no non-constant units
in A.