Abstract
We describe and classify all group gradings on the matrix algebra M 3(k), where k is an arbitrary field. We show that any such grading is either isomorphic to a good grading, for which all the matrix units are homogeneous elements, or reduces to a C 3-grading or to a C 3 × C 3-grading. We show that a grading which is not isomorphic to a good grading is a graded division ring. The isomorphism types of non-good C 3-gradings are in a bijective correspondence to cubic Galois extensions of k. The non-good C 3 × C 3-gradings which do not reduce to C 3-gradings are fine gradings, and their description also depends on cubic Galois extensions of k.
2000 MSC:
ACKNOWLEDGMENT
This work was supported by Kuwait University Research Grant No. SM03/05. The work was also partially supported by the Grant CERES 4-147/2004.
Notes
Communicated by E. R. Puczylowski.