Zero-divisor graphs, von Neumann regular rings, and Boolean algebras

By David F. Anderson, R. Levy, Jay Shapiro. 2003

233 citations recorded by the source corpus.

Abstract For a commutative ring R with set of zero-divisors Z ( R ), the zero-divisor graph of R is Γ ( R )= Z ( R )−{0}, with distinct vertices x and y adjacent if and only if xy =0. In this paper, we show that Γ ( T ( R )) and Γ ( R ) are isomorphic as graphs, where T ( R ) is the total quotient ring of R , and that Γ ( R ) is uniquely complemented if and only if either T ( R ) is von Neumann regular or Γ ( R ) is a star graph. We also investigate which cardinal numbers can arise as orders of equivalence classes (related to annihilator conditions) in a von Neumann regular ring.

Subfields: Algebra and Number Theory, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/819766e6da5568d1724805779a20aa0e475ad3c1

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Citations in the corpus

Commutative rings with toroidal zero-divisor graphs (2007, unknown): “Recently, this subject has been extensively studied in [2], [3], [4], [ 5 ], [7], [8], [12], [13], [15], [18], [19], and [20].”
A Zero Divisor Graph Determined by Equivalence Classes of Zero Divisors (2007, background): “The graph of equivalence classes of zero divisors has also been studied by several others in different contexts, see, for example, [1, 12].”
Zero-Divisor Graphs of Matrices Over Commutative Rings (2009, methodology): “Let T R be the total quotient ring of a commutative ring R. Anderson et al. (2003) showed that R and T R were isomorphic as graphs, and as a consequence it followed that these two graphs had the same diameter and girth.”
Zero-Divisor Graph with Respect to an Ideal (2006, background): “…Iran S. Yassemi Department of Mathematics and Computer Science, Faculty of Science, University of Tehran, Tehran, Iran and School of Mathematics, Institute for Studies in Theoretical Physics and Mathematics, Tehran, Iran LetR be a commutative ring with nonzero identity and let I be an ideal ofR.”
On the Dot Product Graph of a Commutative Ring (2015, unknown): “Recently, there has been considerable attention in the literature to associating graphs with algebraic structures (see [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [17], [18], [19], [20], [21], [23], [24], [25], and [26]).”
On the Zero-Divisor Graph of a Ring (2008, background): “For other articles on zero-divisor graphs, see Anderson and Naseer (1993), Anderson (2008), Anderson et al. (2001, 2003), Anderson and Mulay (2007), Axtel et al. (2005), Axtel and Stickles (2006), DeMeyer and Schneider (2002), Lucas (2006), Mulay (2002), Redmond (2007), and Smith (2003).”