On Commuting Graphs of Finite Matrix Rings

Abstract

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(M _n (F)), for a finite field F and n ≥ 2, then R ≈ M _n (F). Here we prove this conjecture when n = 2.

Key Words:

• Commuting graph
• Finite ring
• Matrix ring

2000 Mathematics Subject Classification:

• 15A33
• 16P10

ACKNOWLEDGMENTS

This research was in part supported by a grant from IPM. The author thanks Professor S. Akbari for motivation.

Notes

Communicated by E. Puczylowski.