| 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 |
| 非交换环的拟零因子图 |
| Quasi-Zero-Divisor Graphs of Non-Commutative Rings |
| 投稿时间：2015-06-05 修订日期：2016-07-29 |
| DOI：10.3770/j.issn:2095-2651.2017.02.002 |
| 中文关键词: 拟零因子 零因子图 着色数 团数 FIC环 |
| 英文关键词:quasi-zero-divisor zero-divisor graph chromatic number clique number FIC ring |
| 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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| 中文摘要: |
| 在本文中,我们引入了一类环FIC环用于研究环的拟零因子图. 令$R$是一个环. 环$R$的拟零因子图,记为$\Gamma_*(R)$, 是一个定义在$R$的非零拟零因子上的有向图,图中的顶点$x$到另一个顶点$y$有一条边当且仅当$xRy=0$. 我们证明了下面三个条件等价: (1) $\chi(R)$是有限的; (2) $\omega(R)$是有限的; (3) $R$的素理想的有限交$\text{Nil}_*(R)$是有限的. 此外,我们还完全决定了图$\Gamma_*(R)$的连通性,直径,围长. |
| 英文摘要: |
| 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 $\Gamma_*(R)$, 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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