5712

5327

كارشناسي ارشد

رياضي محض

اصفهان: دانشگاه صنعتي اصفهان، دانشكده علوم رياضي

1389

نه،95: مصور﴿بخشي رنگي﴾

ص.ع.به فارسي و انگليسي

محمود بهبودي

16/1/90

منصور معتمدي، احمد حقاني

رياضي

ID5327

به فارسي و انگليسي: قابل رويت در نسخه ديجيتالي

The Genus of Zero Divisor and Annihilating Ideal Graphs of Commutative Rings Farid Aliniaeifard f aliniaeifard@math iut ac ir February 15 2011 Master of Science Thesis in Farsi Department of Mathematical Sciences Isfahan University of Technology Isfahan 84156 83111 IranFirst supervisor Dr Mahmood Behboodi mbehbood@cc iut ac irSecond supervisor Dr Hossein Khabazian khabaz@cc iut ac ir2010 MSC Primary 13A99 Secondary 13M05 Key words Commutative ring Zero divisor Annihilating ideal Genus of a graph Abstract Let Si denote the sphere with i handles where i is a nonnegative integer That is Si is anorientable surface of genus i The genus of a graph G denoted G is the minimal integerm such that the graph can be embedded in Sm We say a graph G is toroidal if G 1 Let R be a commutative ring and Z R be the set of all zero divisors of R The zero divisorgraph of R denoted by R is a graph with the vertices set Z R 0 in which two verticesx and y are adjacent if xy 0 In this thesis we present an expanded accounts of toroidalzero divisor graphs based on articles by Cameron Wickham 2008 and 2009 Also we statesome new results we have got through this thesis We investigate the genus of zero divisor graph In particular commutative rings R for which R is toroidal are characterized Also it is shown that for every positive integer g thereare nitely many nite ring with R g Let R be a commutative ring with A R its set of ideals with nonzero annihilator Theannihilating ideal graph of R denoted by AG R is an undirected graph with verticesA R A R 0 in which two vertices I and J are adjacent if IJ 0 It is shown thatif R is an Artinian ring such that AG R then R has nitely many ideals or R m isa Gorenstein ring with maximal ideal m and v dimR m m m2 2 Also for any two integersg 0 and q 0 there are nitely many isomorphism classes of Artinian rings R satisfyingthe conditions i AG R g and ii R m q for every maximal ideal m of R Also it is shown that if R is a non domain Noetherian local ring such that AG R theneither R is a Gorenstein ring or R is an Artinian ring with nitely many ideals Also we study the interplay between the diameter and girth of annihilating ideal graphs andzero divisor graphs It is shown that if R is a non domain Noetherian ring with AG R K2 then diam AG R diam AG R x diam AG R x1 x2 xn diam AG R x diam R diam R x diam R x1 x2 xn diam R x Also it is shownthat gr R x gr R x gr AG R x gr AG R x and in addition if Ris a reduced ring with AG R and R contain cycles then gr R gr R x gr R x gr AG R gr AG R x gr AG R x Finally we extend the notionof the annihilating ideal graph to non commutative rings 1

محمود بهبودي

منصور معتمدي، احمد حقاني