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R（+）M与R（？）I的零因子图
The Zero-divisor Graphs of R(+)M and R(?)I
【作者】 吴中伟；
【导师】 邓培民；
【作者基本信息】 广西师范大学 ， 基础数学， 2009， 硕士
【摘要】 有限交换环一直是代数学中的不可或缺的重要研究对象,它不仅内涵丰富,而且在众多数学分支(如:组合数学,有限几何和算法分析)以及工程科学(如:编码理论和序列设计)中有着重要的应用.特别是最近二三十年中,由于有限环在编码领域中的作用越来越突出,使得对有限环的研究成果更加丰富.用图的性质研究代数结构是最近20年来才产生的一个新型研究领域,引发出了很多有趣的结果和问题.它主要涉及数学中的六个领域:环论,群论,半群论,图论,初等数论和组合数学,如此众多的交叉研究,使它不但具有趣味性和吸引力,而且有广阔的发展前景.近年,它已成为国际上的一个热门研究领域.在本文的第一章中我们概述了零因子图发展的历史,本文的研究背景以及本文的主要结果.同时我们还给出了环论与图论的一些基本的概念和结论.第二章中我们研究了任意交换环的idealization的零因子图的平面性,得到了一些较满意的结果.在本文的第三章中,我们讨论了idealization的零因子图的围长.对有限交换环的idealization的零因子图的围长给出了比[28]更具体的刻划.在第四章中我们研究了amalgamated duplication of a commutative ring along an ideal的零因子图的平面性,得到了一些较满意的结果,且讨论了它的零因子图的直径.在本文的第五章中我们讨论了交换环的idealization的基于理想的零因子图.特别地,我们刻划了它的零因子图的围长与直径.
【Abstract】 The theory of finite commutative rings is a very active area which is not only of great theoreticalinterest in itself but also found important applications both within mathematics (for instance,in Combinatorics, Finite Geometries and the Analysis Algorithms) and within the EngineeringSciences (in particular in Coding Theory and Sequence Design). especially,in recent decades,there are more and more abundant research results of finite ring because of the finite ring play amore and more prominent role in coding areas. The study of algebraic structures, using properties ofgraphs, has become an exciting research topic in the last twenty years, leading to many fascinatingresults and questions. It lies at the crossroads of six areas: ring theory, group theory, semigrouptheory, graph theory, elementary number theory and combinatorics. Like so many interdisciplinarystudies, it has its fascinations, attractions, and also vast potential for future development. It hasbecome an active research topic in recent years.In Chapter 1 of this paper, we summarize the history of the zero-divisor graphs, and give thebackground and main results of this paper. At the same time, we give the notation and basicresults of ring theory and graph theory.In Chapter 2, we will determine the planarity of the zero-divisor graph of idealization of anycommutative ring and get some approving results. The main results of this chapter will be veryuseful in the following chapters.In Chapter 3, we will study the girth of the zero-divisor graph of idealization, give more specificcharacterizations about the girth of the zero-divisor graph of idealization of any finite commutativerings than those given in [28].In Chapter 4, we will determine the planarity of the zero-divisor graph of amalgamated du-plication of a commutative ring along an ideal and get some satisfying results, and also study thediameter of the zero-divisor graph.In Chapter 5, we will discuss ideal-based zero-divisor graphs of idealization of any commutativerings. Specifically, we characterize the girth and diameter of the graphs.