有限交换环的代数结构与图结构

Recently, the concept of zero-divisor graphs provide an important idea for studying the structures and classifications of commutative rings, and attract research interests of many algebraists. By classifying the graphic structures of some zero-divisor graphs, we launched a series of research about the algebraic structures and isomorphic classifications of finite commutative rings in our previous work. Firstly, based on the previous work, this project is about to go on studying the corresponding relationship between zero-divisor graphs with more complicated structures (i.e. regular graph, complete graph with many horns) and finite commutative rings. Secondly, by the classification of the graphic structure of refinements of a star graph, we give the algebraic characterization of the minimal generating set of the maximal ideal of finite commutative local rings, and hence determine the algebraic structure of finite commutative local rings under isomorphism. Finally, drawing on on the existing research method for automorphism groups of graphs, by characterizing some vertex subsets of graphs and applying the methods of graph theory and group theory, we determine the automorphism groups of some zero-divisor graphs. By the work of this research project, we are making efforts to reveal more specific structure of the objects related to zero-divisor graphs defined on finite commutative rings, enrich the structure theories of finite commutative local rings, and characterize more automorphism groups of zero-divisor graphs.

近年来，借助零因子图研究交换环的结构与分类是代数学研究中的热点问题。本项目组在前期工作中通过对零因子图结构的划分，对有限交换环的代数结构与同构分类展开了一系列研究。首先，本项目拟在前期研究的基础上，继续对一些结构较为复杂的图类型（如正则图、完全图带多个角）展开图与有限交换环之间对应关系的研究；然后，通过对星图加细结构的分类讨论，给出有限交换局部环的极大理想的极小生成元集的代数刻画，进而确定有限交换局部环的代数结构与同构分类。最后，借鉴已有的图自同构群的研究方法，通过对图中某些顶点子集的完全刻画，运用图论和群论的方法来刻画部分零因子图的自同构群。通过本项目的研究，我们希望结合图论的直观性、组合的技巧性与代数的深刻性，挖掘出更多的交换环上零因子图的具体结构类型，进一步丰富和完善有限交换局部环的分类结构定理，刻画出更多的有限交换环上零因子图的自同构群。

图论与代数的交叉研究是代数学研究中的热点问题。本项目借助有限交换环的零因子图结构，通过对图结构的分类讨论，刻画了团数是3，4，5的零因子图对应的所有有限交换环的代数结构。特别的，我们将阶小于或等于81的有限局部环的代数结构及同构分类完全确定下来。最后，借鉴已有的图自同构群的研究方法，我们确定了整环的零因子图和独点集的自同构群。本项目的研究挖掘出更多的交换环上零因子图的具体结构类型，进一步丰富和完善有限交换局部环的分类结构定理。