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.
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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.