| 非交换环的拟零因子图 |
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| 引用本文： | 赵寿祥,南基洙,唐高华.非交换环的拟零因子图[J].数学研究及应用,2017,37(2):137-147. |
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| 作者姓名： | 赵寿祥 南基洙 唐高华 |
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| 作者单位： | 大连理工大学数学科学学院, 辽宁 大连 116024,大连理工大学数学科学学院, 辽宁 大连 116024; 桂林师范高等专科学校数学与计算机技术系, 广西 桂林 541001,广西师范学院数学科学学院, 广西 南宁 530023 |
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| 基金项目： | 国家自然科学基金(Grant Nos.11371343; 11161006; 11661014; 11171142), 广西科学研究与技术开发计划项目(Grant No.1599005-2-13),广西高校科学技术研究项目(Grant No.KY2015ZD075), 广西自然科学基金(Grant No.2016GXSFDA380017). |
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| 摘 要： | 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 Γ_*(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 x Ry = 0. We show that the following three conditions on an FIC ring R are equivalent:(1) χ(R) is finite;(2) ω(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 Γ_*(R).
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| 关 键 词： | 拟零因子 零因子图 着色数 团数 FIC环 |
| 收稿时间： | 2015/6/5 0:00:00 |
| 修稿时间： | 2016/7/29 0:00:00 |
Quasi-Zero-Divisor Graphs of Non-Commutative Rings |
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| Shouxiang ZHAO,Jizhu NAN and Gaohua TANG.Quasi-Zero-Divisor Graphs of Non-Commutative Rings[J].Journal of Mathematical Research with Applications,2017,37(2):137-147. |
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| Authors: | Shouxiang ZHAO Jizhu NAN and Gaohua TANG |
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| Institution: | 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,School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China; College of Sciences, Shenyang Agricultural University, Liaoning 110866, P. R. China and 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 $\Gamma_*(R)$. |
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| Keywords: | quasi-zero-divisor zero-divisor graph chromatic number clique number FIC ring |
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