Abelian Group Gradings on Full Matrix Rings

Abstract

For positive integers ℓ and n, several authors studied ℤ_ℓ-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).

Key Words:

• Graded algebra
• Matrix algebra

Mathematics Subject Classification:

• Primary 16W50
• Secondary 16S50

ACKNOWLEDGMENT

The authors would like to express their gratitude to the referee for valuable comments and constructive suggestions of the manuscript.

Jaeun Lee is supported by Com²MaC-KOSEF (R11-1999-054).

Notes

Communicated by M. Ferrero.