Zero-divisor Graphs Of Group Rings Of The Symmetric Group On Three Letters

The study of the zero-divisor graphs are related to many areas:ring theory,group theory, semigroup theory,field theory,graph theory and elementary number theory. So many subjects intercross,make it fascinated and attracted,but also inherent dilemmas.The study of algebraic structures,using properties of graphs,has become an exciting research topic in the last twenty years,leading to many fascinating results and questions.This paper is composed of five parts,where the first part is the introduction,the second to the forth in which each part is a chapter are the core of the paper,and the last part is the further research problems.In Chapter 1 of this paper,we summarize the history and current situation of the zero-divisor graph,the research background of this paper.At the same time,we gave some notations of ring theory and graph theory.In Chapter 2,we will determine the zero-divisors of ZnS3,and discuss the girth and diameter of the directed zero-divisor graphΓ(ZnS3).The main result has been published in Journal of Guangxi Normal University 12(4)(2010).These results will be very useful in the following chapters,and they are as following:Theorem 2.3 Let Zn be the modulo n residue class ring,S3 be the symmetric group on three letters,R=ZnS3,then the zero-divisors of R are(1)If n=2,then D(R)={α∈R││supp(α)│=0,2,3,4,6},│D(R)│=52;(2)If n=3,then D(R)={α=x1+x2b+x3ab+x4a2b+x5a+x6a2│xi∈Z3,x1+x5+x6≡x2+x3+x4(mod 3),or (?)xi≡0(mod 3)},│D(R)│=405;(3)If n=p,p>3 is a prime number,then ZpS3≌Zp⊕Zp⊕M2(Zp),and D(Zp⊕Zp⊕M2(Zp))-{(α1,α2(?))│at least one ofα1 andα│D(ZpS3)│=3p5-2p4-2p3+3p2-p;(4)If n=pt,t>1,then ZptS3/IS3≌ZpS3.D(ZptS3=D(ZpS3)+IS3,in which I={0,p,2p,…,(pt-1-1)p}is the unique maximal ideal of Zpt;(5)If n=p1r1p2r2…p3r3,ri≥1,s≥2,then R≌Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3, D(Zp1riS3⊕Zp2r2S3⊕…⊕Zp3r3S3)={(α1,α2,…,αs)│αi∈ZpiriS3,at least one ofαi∈D(ZpiriS3),i=1,2,3,…,s}. Theorem 2.4 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3,R=ZnS3,n>1, then(1) diam{Γ(R))=2(?)n=3t,t≥1;(2) diam(Γ(R))=3(?)n≠3t,t≥1.Theorem 2.5 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, then gr(Γ(ZnS3))=3.Theorem 2.6 Let Zn be the modulo n residue class ring, S3 be the symmetric group on three letters, R=ZnS3, n>1, thenΓ(ZnS3) is non-planar.In Chapter 3, we generalize the residue class ring to arbitrary finite ring R, we determine the algebraic proposition and structure of RS3, and discuss the girth and diameter of the directed zero-divisor graphΓ(RS3). The main results are:Theorem 3.4 Let R be a finite ring of characteristic pT, S3 be the symmetric group on three letters, if (p,2)=(p,3)=1, then diam(Γ(RS3))=3.Theorem 3.6 Let R be a finite ring, S3 be the symmetric group on three letters, then gr(r(RS3))=3.Theorem 3.7 Let R be a finite ring, S3 be the symmetric group on three letters, thenΓ(RS3) is non-planar.In Chapter 4, we generalize the symmetric group on three letters to m letters, when p│m!, we make some meaningful exploration of ZpSm, and discuss the girth and planarity. The main results are:Theorem 4.2 Let Zp be the modulo p residue class ring, Sm be the symmetric group on m letters, if p│m!, then ZpSm≌Zp⊕Zp⊕△(Sm, Am)Theorem 4.4 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, then gr(Γ(ZnAm))=3.Theorem 4.5 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, then gr(Γ(ZnSm))=3.Theorem 4.7 Let Zn be the modulo n residue class ring, Am be the alternative group on m letters, if m≥4, thenΓ(ZnAm)) is non-planar.Theorem 4.8 Let Zn be the modulo n residue class ring, Sm be the symmetric group on m letters, if m≥3, thenΓ(ZnSm) is non-planar.