QuasiZeroDivisor Graphs of NonCommutative Rings 
Received:June 05, 2015 Revised:July 29, 2016 
Key Word:
quasizerodivisor zerodivisor graph chromatic number clique number FIC ring

Fund ProjectL:Supported by the National Natural Science Foundation of China (Grant Nos.11371343; 11161006; 11661014; 11171142), the Guangxi Science Research and Technology Development Project (Grant No.1599005213), the Scientic Research Fund of Guangxi Education Department (Grant No.KY2015ZD075) and the Natural Science Foundation of Guangxi (Grant No.2016GXSFDA380017). 
Author Name  Affiliation  Shouxiang ZHAO  School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China College of Sciences, Shenyang Agricultural University, Liaoning 110866, P. R. China Department of Mathematics and Computer Science, Guilin Normal College, Guangxi 541001, P. R. China  Jizhu NAN  School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China College of Sciences, Shenyang Agricultural University, Liaoning 110866, P. R. China  Gaohua TANG  School of Mathematical and Statistics Sciences, Guangxi Teachers Education University, Guangxi 530023, P. R. China 

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Abstract: 
In this paper, a new class of rings, called FIC rings, is introduced for studying quasizerodivisor graphs of rings. Let $R$ be a ring. The quasizerodivisor graph of $R$, denoted by $\Gamma_*(R)$, is a directed graph defined on its nonzero quasizerodivisors, where there is an arc from a vertex $x$ to another vertex $y$ if and only if $xRy=0$. We show that the following three conditions on an FIC ring $R$ are equivalent: (1) $\chi(R)$ is finite; (2) $\omega(R)$ is finite; (3) Nil$_*R$ is finite where Nil$_*R$ equals the finite intersection of prime ideals. Furthermore, we also completely determine the connectedness, the diameter and the girth of $\Gamma_*(R)$. 
Citation: 
DOI:10.3770/j.issn:20952651.2017.02.002 
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