Group gradings on full matrix rings

We study G-gradings of the matrix ring M-n(k), k a field, and give a comple te description of the gradings where all the elements e(i,j) are homogeneou s, 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 p roduced. 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 isomo rphic to one of the two good gradings. (C) 1999 Academic Press.