Group gradings on full matrix rings

We study G-gradings of the matrix ring M-n(k), k a field, and give a complete description of the gradings where all the elements e(i,j) are homogeneous, called good gradings. Among these, we determine the ones that are strong gradings or crossed products. If G is a finite cyclic group and k contains a primitive \G\th root of 1, we show how all G-gradings of M-n(k) can be produced. In particular we give a precise description of all C-2-gradings of M-2(k) and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings. (C) 1999 Academic Press.