Graded monomial identities and almost non-degenerate gradings on matrices

Let F be a field of characteristic zero, G be a group and be the algebra of matrices of size n with entries from F with a G-grading. Bahturin and Drensky proved that if the G grading on is elementary and the neutral component of is commutative, then the graded identities of follow from three basic types of identities and monomial identities of length ≥2 bounded by a function of n. In this paper we prove the best upper bound is . More generally, we prove that all the graded monomial identities of an elementary G-grading on follow from those of degree at most n. We also study gradings which satisfy no graded multilinear monomial identities but the trivial ones, which we call almost non-degenerate gradings. The description of non-degenerate elementary gradings on matrix algebras is reduced to the description of non-degenerate elementary gradings on matrix algebras that have commutative neutral component. We provide necessary conditions so that the grading on is almost non-degenerate and we apply the results on monomial identities to describe all almost non-degenerate -gradings on M_n(F) for n<=5.