A note on commuting graphs of matrix rings over fields

TL;DR

This paper provides a concise proof that for fields with finite algebraic closure, the commuting graph of matrix rings over these fields is connected with a diameter of four when matrix size is at least three.

Contribution

It offers a simplified proof of the connectivity and diameter of commuting graphs for matrix rings over certain fields, extending understanding of their algebraic structure.

Findings

Commuting graph is connected for n ≥ 3 over fields with finite algebraic closure.

Diameter of the commuting graph is exactly four in these cases.

Proof simplifies previous results on commuting graphs of matrix rings.

Abstract

We will give a short proof of the fact that if the algebraic closure of a field $\mathbb F$ is a finite extension, then for $n\geq 3$ the commuting graph $\Gamma(M_n(\mathbb F))$ is connected and its diameter is four.