非交换环的拟零因子图

Quasi-Zero-Divisor Graphs of Non-Commutative Rings

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

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 $\Gamma_*(R)$.