Zero-divisor graphs, von Neumann regular rings, and Boolean algebras |
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Authors: | David F Anderson |
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Institution: | a Department of Mathematics, The University of Tennessee, Knoxville, TN 37996, USA b Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030, USA |
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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. |
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Keywords: | Primary: 13A99 06E99 secondary: 13M99 |
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