模的交图
Intersection Graph of a Module
摘要：
设$V$是一个左$R$ -模，其中$R$是一个（不一定是交换的）有单位环。$V$的真$R$ -子模的交图$\cG(V)$是一个没有环和多重边的无向图，定义如下：顶点集是$V,$的所有真$R$ -子模的集合，并且在两个不同的顶点$U$和$W$之间存在边当且仅当$U\cap W\neq 0.$我们研究这些图以将$\cG(V)$的组合性质与$R$的代数性质联系起来，我们研究了$V.$的连通性、控制性、有限性、染色性和平面性。例如，我们找到了$\cG (V).$的控制数，我们还在某些情况下找到了$\cG(V)$的色数。此外，我们研究了$\cG(V),$中的圈和$\cG (V)$中的完全子图，确定了$\cG(V)$是平面的$V$的结构。
英文摘要：
Let $V$ be a left $R$-module where $R$ is a (not necessarily commutative) ring with unit. The intersection graph $\cG(V)$ of proper $R$-submodules of $V$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper $R$-submodules of $V,$ and there is an edge between two distinct vertices $U$ and $W$ if and only if $U\cap W\neq 0.$ We study these graphs to relate the combinatorial properties of $\cG(V)$ to the algebraic properties of the $R$-module $V.$ We study connectedness, domination, finiteness, coloring, and planarity for $\cG (V).$ For instance, we find the domination number of $\cG (V).$ We also find the chromatic number of $\cG(V)$ in some cases. Furthermore, we study cycles in $\cG(V),$ and complete subgraphs in $\cG (V)$ determining the structure of $V$ for which $\cG(V)$ is planar.
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