Zero-divisor Graphs of Matrices over Commutative Rings

The concept of a zero-divisor graph of a commutative ring was first introduced in Beck (1988), and later redefined in Anderson and Livingston (1999). Redmond (2002) further extended this concept to the noncommutative case, introducing several definitions of a zero-divisor graph of a noncommutative ring. Recently, the diameter and girth of polynomial and power series rings over a commutative ring were studied in Axtell et al. (2005), Anderson and Mulay (2007), and Lucas (2006). Let R be a commutative ring with 1, and let Mn!R" denote the ring of n× n matrices over R. Let #!R" and #!Mn!R"" denote the zero-divisor graphs of R and Mn!R", respectively. The object of this article is to find a relation between the diameters of #!R" and #!Mn!R"". We view this problem as a natural continuation of the investigation into relations between diameters of zero-divisor graphs of a commutative ring R and polynomials and power series over the same ring. Unlike these results, our case involves an extension of a commutative ring into a noncommutative one and, therefore, relates graphs of commutative and noncommutative rings. In Section 2, we recollect the main definitions and the results which we need for both the commutative and noncommutative case. In Section 3, we prove several theorems about zero-divisor graphs of matrix rings. In Section 4, we turn to our main problem—the investigation of possible diameters of #!Mn!R"" in terms of the diameter of #!R".