Induced good gradings of structural matrix rings

Abstract

Our approach to structural matrix rings defines them over preordered directed graphs. A grading of a structural matrix ring is called a good grading if its standard unit matrices are homogeneous. For a group G, a G-grading set is a set of arrows with the property that any assignment of these arrows to elements of G uniquely determines an induced good grading. One of our main results is that a G-grading set exists for any transitive directed graph if G is a group of prime order. This extends a result of Kelarev. However, an example of Molli Jones shows there are directed graphs which do not have G-grading sets for any cyclic group G of even order greater than 2. Finally, we count the number of nonequivalent elementary gradings by a finite group of a full matrix ring over an arbitrary field.

Keywords:

• Graded algebra
• directed graph

2000 MATHEMATICS SUBJECT CLASSIFICATION:

• 16W50 (Primary)
• 16S50 (Secondary)

Acknowledegements

The authors express their gratitude to the referee. One of the authors, Price, dedicates his contributions to Dr. Jeffrey Bergen, who recently passed away.