The coordinate ring $O_q(M_n)$, for all matrices, is an example of a multivariate "skew polynomial ring, which means a graded or filtered ring with certain axioms that imply that any map from a vector of exponents to monomials is a basis. (I am getting this from Noncommutative Gr??bner bases and filtered-graded transfer, by Huishi Li.) In any ring like this, there is a theory of Gr??bner bases