Induced Good Gradings of Structural Matrix Rings

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.