journal6 ›› 2016, Vol. 37 ›› Issue (1): 11-13.DOI: 10.3969/j.cnki.jdxb.2016.01.003

• Mathematics • Previous Articles     Next Articles

Zero-Divisor Graph of Von Neumann Regular Rings

 PAN  Hong-Yan, LI  Ai-Hua   

  1. (College of Mathematics and Statistics,Jishou University,Jishou 416000,Hunan China)
  • Online:2016-01-25 Published:2016-01-21

Abstract: In this paper we investigate the zero-divisor graph of von Neumann regular rings,and we focus our main attention on the property of vertices and the connectedness of the zero-divisor graph Γ(R).Let R be a von Neumann regular ring with unit 1,we show that  Γ(R) is connected if and only if R is direct finite.In addition,if R is a regular ring without unity elements,then  Γ(R) is connected if and only if R has no proper one-sided identity elements.For the zero-divisor graph  Γ(R) of a regular ring R with |R|≥ 5,we have that Sour(R)={a∈R|a is right invertible but  not left invertible} and Sink (R)={a∈R|a is left invertible but not right invertible}.

Key words: Von Neumann regular rings, zero-divisor graphs, connectedness, sources, sinks

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