Group Gradings on Full Matrix Rings

We study G-gradings of the matrix ring Mn(k), k a field, and give a complete description of the gradings where all the elements ei, 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 Gth root of 1, we show how all G-gradings of Mn(k) can be produced. In particular we give a precise description of all C2-gradings of M2(k) and show that for algebraically closed k, any such grading is isomorphic to one of the two good gradings.