Commuting graphs of full matrix rings over finite fields. (Q924367)

For a non-commutative ring \(R\), the commuting graph \(\Gamma(R)\) of \(R\) is the undirected graph with vertex set the elements of \(R\) which are not in its center. In this graph, two distinct vertices are joined by an edge if they commute with each other. It has been conjectured that for a ring \(R\), a finite field \(F\) and an integer \(n\geq 2\), if \(\Gamma(R)\cong\Gamma(M_n(F))\), then \(R\cong M_n(F)\). Here \(M_n(F)\) denotes the full \(n\times n\) matrix ring over \(F\). The author investigates the validity of this conjecture. Although the last word on this conjecture is still to be spoken, the author has made a significant contribution in support of the conjecture. In particular, it is shown that for any finite field \(F\), \(n\geq 2\) and \(R\) a ring with identity with \(\Gamma(R)\cong\Gamma(M_n(F))\), the rings \(R\) and \(M_n(F)\) must have the same cardinality. If the cardinality of the field \(F\) is prime and \(n=2\), then \(R\cong M_2(F)\). Using truncated skew-polynomial rings, the author shows that isomorphic commuting graphs is in general not sufficient to ensure that the corresponding rings have the same cardinality.