交换环的二部本质图

Bipartite Essential Graphs of Commutative Rings

交换环R的本质图EG(R)是一个无向简单图,它以Z(R)\{0}为顶点集,两个不同的顶点x、y之间有一条边相连当且仅当ann(xy)是R的一个本质理想.给出了模n剩余类环Zn的零因子图与本质图相等的充分必要条件.在此基础上,证明了交换环的二部本质图必是完全二部图,并对相应的环进行了同构分类.

For a commutative ringR, its essential graphEG(R) is an undirected simple graph whose vertex set is Z(R) \{0}, and two distinct verticesxandyare adjacent if and only if ann(xy) is an essential ideal. By giving a necessary and sufficient condition forZnsuch that its zero-divisor graph coincides with its essential graph, it is showed that a bipartite essential graph of a commutative ring must be a complete bipartite graph, and the classifications of the corresponding rings up to isomorphism are also established.