The Properties Of Zn[i] And Zero-Divisor Semigroups Of Some Simple Graphs

The subject matter of this paper lies at the crossroads of six areas: ring the-ory, group theory, semigroup theory, graph theory, elementary number theory andcombinatorics. Like so many interdisciplinary studies, it has its fascinations andattractions, but also its inherent dilemmas.The theory of finite commutative rings is very active area which is not only ofgreat theoretical interest in itself but also found important applications both withinmathematics(for instance, in Combinatorics, Finite Geometries and the AnalysisAlgorithms) and within the Engineering Sciences(in particular in Coding Theoryand Sequence Design). In fact, several codes over finite fields which are widelyused in Information and Communication Theory are best understood as images ofcodes over Galois rings(especially over the ring of integers modulo 4). The studyof algebraic structures, using properties of graphs, has become an exciting researchtopic in the last twenty years, leading to many fascinating results and questions.In Chapter 1 of this paper, we summarize the history of the zero-divisor graph,the background and main results of this paper. At the same time, we gave thenotation and basic results of ring theory, graph theory and semigroup theory.In Chapter 2, we will determine the prime spectrum, zero-divisors and units ofZn[i]. The main results of this chapter will be very useful in the following chap-ters. The main result has been published in Journal of Guangxi Teachers'College23(2006).In 1801, Gauss proved that the structure theorem of the unit group U(Zn) ofthe residue class ring Zn. In Chapter 3, we will prove the structure theorem ofthe unit group U(Zn[i]) of the Gaussian integers modulo n. The main results have contributed in Journal of Mathematical Research and Exposition. This paper hasalready passed first trial and the referer's report are very positive.In Chapter 4, the properties of the zero-divisor graphΓ(Zn[i]) of the Gaussianintegers modulo n are investigated, including the diameter of graph , the planarity,the girth of graph and the center. The main results have been accepted and will bepublished in Journal of Guangxi Normal University. The main results of Chapter2,3,4 have been reported at the tenth national conference on algebra in 2006, it getaltitudinal attention for many scholars.In chapter 5, we investigated zero-divisor semigroups and commutative rings ofa series of simple graphs. We obtained a lot of results.First, in section 5.2, we determine all commutative zero-divisor semigroups ofall simple graphs with at most four vertices. The main results of this section havecontributed in Algebra Colloquium. This paper has already passed first trial andthe referer's report are very positive.In section 5.3, we completely determine the commutative rings and commutativezero-divisor semigroups of regular polyhedrons.The main result of section 5.4 is that we obtain a formula H(n) to calculatethe number of non-isomorphic zero-divisor semigroups corresponding to the graphKn?e, the complete graph Kn deleted an edge, and by using of a computer program,the values of H(n) are listed for 1≤n≤100.In section 5.5, we proved that there are no commutative rings corresponding tothe graph Km,n,p + v, for any m,n,p, and there exist no commutative zero-divisorsemigroup corresponding to the graph Km,n,p + v, for any m,n,p≥2, and thereare commutative zero-divisor semigroups corresponding to the graph Km,n,p + v ifm,n,p≥1 and one of m,n,p is 1. We determine zero-divisor semigroups of twoespecial cases of the graph Km,n,1 + v. Furthermore, we determine all commutativezero-divisor semigroups of the graph Km,n,1 + v, for n = 1, 2, 3.In section 5.6 of this paper, we give formulas to calculate the numbers ofnon-isomorphic zero-divisor semigroups corresponding to star graphs K1,n, two-stargraphs Tm,n and windmill graphs respectively.