论文标题
零局部图的诱导子图
Induced subgraphs of zero-divisor graphs
论文作者
论文摘要
具有Unity的有限交换环的零分数图是其顶点集是环中的零偏差集，如果$ ab = 0 $，则$ a $ a $ a $ a $ b $。我们表明，零分类的类别是通用的，因为每个有限图都是归因于零径移图的诱导子图的同构。对于包括布尔环，田野和当地戒指在内的各种限制类戒指，这仍然是正确的。但是在受限制的类别中，零分量图不会形成通用家庭。例如，局部环的零分量图，其最大理想是主体是阈值图。并且每个阈值图都可以嵌入在此环的零分离器图中。更普遍地，我们在非本地环上提供了必要和充分的条件，其零分离器图作为阈值图。此外，我们表明有一个可数的本地环，其零分量图嵌入了rado图，因此，每个有限或可计数的图（如诱导的子图）。最后，我们考虑在相关图中的嵌入，例如$ 2 $维点产品图。
The zero-divisor graph of a finite commutative ring with unity is the graph whose vertex set is the set of zero-divisors in the ring, with $a$ and $b$ adjacent if $ab=0$. We show that the class of zero-divisor graphs is universal, in the sense that every finite graph is isomorphic to an induced subgraph of a zero-divisor graph. This remains true for various restricted classes of rings, including boolean rings, products of fields, and local rings. But in more restricted classes, the zero-divisor graphs do not form a universal family. For example, the zero-divisor graph of a local ring whose maximal ideal is principal is a threshold graph; and every threshold graph is embeddable in the zero-divisor graph of such a ring. More generally, we give necessary and sufficient conditions on a non-local ring for which its zero-divisor graph to be a threshold graph. In addition, we show that there is a countable local ring whose zero-divisor graph embeds the Rado graph, and hence every finite or countable graph, as induced subgraph. Finally, we consider embeddings in related graphs such as the $2$-dimensional dot product graph.
