Quasi-Zero-Divisor Graphs of Non-Commutative Rings
Received:June 05, 2015  Revised:July 29, 2016
Key Word: quasi-zero-divisor   zero-divisor 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.1599005-2-13), 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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In this paper, a new class of rings, called FIC rings, is introduced for studying quasi-zero-divisor graphs of rings. Let $R$ be a ring. The quasi-zero-divisor graph of $R$, denoted by $\Gamma_*(R)$, is a directed graph defined on its nonzero quasi-zero-divisors, 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 $\Gamma_*(R)$.