The commuting graph of the ring M₃(F_q)

Abstract

Let R be a noncommutative ring with unity and $Z(R)$ be its centre. The commuting graph of R denoted by $\mathrm{\Gamma }(R)$ is a graph whose vertices are noncentral elements of R and two distinct vertices x and y are adjacent if and only if xy = yx. Let F be a finite field. In this paper, we show that if $\mathrm{\Gamma }(R)\cong \mathrm{\Gamma }(M_3(F))$ and $Z(R)$ is a field, then $R\cong M_3(F)$. In particular, if $\mathrm{\Gamma }(R)\cong \mathrm{\Gamma }(M_3(F_p))$, then $R\cong M_3(F_p)$.

Keywords:

• Commuting graph
• finite field
• centralizer

MSC:

• Primary: 05C50
• 16K20

Acknowledgments

The authors would like to express their deep gratitude to referees for a very careful reading of the paper, and many valuable comments which have greatly improved the presentation of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.