| Quasi-Zero-Divisor Graphs of Non-Commutative Rings |
| Received:June 05, 2015 Revised:July 29, 2016 |
| Key Words:
quasi-zero-divisor zero-divisor graph chromatic number clique number FIC ring
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| Fund Project: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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| Abstract: |
| 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 $\Gamma_*(R)$. |
| Citation: |
| DOI:10.3770/j.issn:2095-2651.2017.02.002 |
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