Abelian Group Gradings on Full Matrix Rings

Abstract

For positive integers ℓ and n, several authors studied ℤℓ-gradings of the full matrix ring M n (k) over a field k. In this article, we show that every (G × H)-grading of M n (k) can be constructed by a pair of compatible G-grading and H-grading of M n (k), where G and H are any finite groups. When G and H are finite cyclic groups, we characterize all (G × H)-gradings which are isomorphic to a good grading. Moreover, the results can be generalized for any finite abelian group grading of M n (k).