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.