Compressed commuting graphs of matrix rings

TL;DR

This paper introduces a new compressed commuting graph for rings that captures commutativity relations across all homomorphic images, with specific computations for finite fields and matrix rings.

Contribution

It defines a compressed commuting graph that induces a functor from rings to graphs, preserving commutativity relations in all homomorphic images, and demonstrates its optimality for matrix algebras over finite fields.

Findings

Computed compressed commuting graphs for finite fields.

Computed compressed commuting graphs for 2x2 matrix rings over finite fields.

Established the optimality of the compression for matrix algebras.

Abstract

In this paper we introduce compressed commuting graph of rings. It can be seen as a compression of the standard commuting graph (with the central elements added) where we identify the vertices that generate the same subring. The compression is chosen in such a way that it induces a functor from the category of rings to the category of graphs, which means that our graph takes into account not only the commutativity relation in the ring, but also the commutativity relation in all of its homomorphic images. Furthermore, we show that this compression is best possible for matrix algebras over finite fields, i.e., it compresses as much as possible while still inducing a functor. We compute the compressed commuting graphs of finite fields and rings of $2 \times 2$ matrices over finite fields.