一些环的Nil-Clean图的导出子图中的完备码

Perfect Codes in Induced Subgraphs of Nil-Clean Graphs for Some Rings

摘要: 设$R$ 是一个环。如果一个元素$r\in R$ 可以分解成$R$ 中的一个幂等元与一个幂零元的和，那么就称$r$ 为$R$ 中的一个nil-clean元。$R$ 的nil-clean图记为$G_{NC}\left(R\right)$ ，其点集为$R$ ，两个不同的点$x$ 和$y$ 是邻接的当且仅当$x+y$ 是$R$ 中的一个nil-clean元。设$G$ 是一个有限无向简单图，其点集为$V\left(G\right)$ 。设$C$ 是$V\left(G\right)$ 的一个非空子集。如果当$c_i$ 取遍$C$ 中所有元素时，$c_i$ 的半径为1的闭邻域$S_1\left(c_i\right)$ 构成$V\left(G\right)$ 的一个划分，则称$C$ 为图$G$ 中的一个完备码。本文通过分析一些环的nil-clean图的导出子图的结构来确定这些图中完备码的大小，其中导出子图的点集为环的单位集。

Abstract: Let $R$ be a ring. If an element $r\in R$ can be decomposed into the sum of an idempotent element and a nilpotent element in $R$ , then $r$ is called a nil-clean element in $R$ . The nil-clean graph of $R$ is denoted as $G_{NC}\left(R\right)$ , whose point set is $R$ , and two different points $x$ and $y$ are adjacent if and only if $x+y$ is a nil-clean element of $R$ . Let $G$ be a finite undirected simple graph with vertex set $V\left(G\right)$ . Let $C$ be a non-empty subset of $V\left(G\right)$ . When $c_i$ runs through $C$ , if $S_1\left(c_i\right)$ form a partition of $V\left(G\right)$ , where $S_1\left(c_i\right)$ is the closed neighbourhood of $c_i$ with radius 1, then $C$ is called a perfect code in $G$ . In this paper, we determine the sizes of the perfect codes in the induced subgraphs of the nil-clean graphs of some rings by analyzing the structure of these graphs, where the vertex sets of the induced subgraphs are the unit sets of the rings.