Анализа прстена и модула придруживањем графова

Abstract

Ova doktorska disertacija prouqava razliqite osobine komutativnihprstena i modula algebarsko kombinatornim metodama. Ako se graf naodgovarajui naqin pridrui prstenu R ili R-modulu M, onda ispitivaem egovih osobina dolazimo do korisnih informacija o R i M.U ovoj tezi odreen je radijus totalnog grafa komutativnog prstenaR u sluqaju kada je taj graf povezan. Tipiqna raxirea kao xto suprsten polinoma, prsten formalnih redova, idealizacija R-modula Mi prsten matrica Mn(R) takoe su ispitani. Ustanov ene su veze izmeutotalnog grafa polaznog prstena R i totalnih grafova ovih raxirea.Definisaem totalnog grafa modula dato je jedno uopxtee totalnoggrafa komutativnog prstena. Ispitane su i dokazane egove razliqiteosobine. Ustanov ene su veze sa totalnim grafom prstena kao i nekeveze sa grafom delite a nule.U ci u bo eg razumevaa qistih prstena, uveden je qisti graf C¡(R)komutativnog prstena sa jedinicom R. Deta no su ispitane egoveosobine. Da im istraivaem qistih grafova dobijeni su dodatnirezultati vezani za druge klase komutativnih prstena.Jedan od predmeta ove teze je i istraivae osobina odgovarajueglinijskog grafa L(T¡(R)) totalnog grafa T¡(R). Data je kompletnaklasifikacija svih komutativnih prstena qiji su linijski grafovi totalnoggrafa planarni ili toroidalni. Dokazano je da za ceo brojg ¸ 0 postoji samo konaqno mnogo komutativnih prstena takvih da je°(L(T¡(R))) = g.U ovoj tezi su takoe klasifikovani svi toroidalni grafovi kojisu grafovi preseka ideala komutativnog prstena R. Dato je i jednopobo xae postojeih rezultata o planarnosti ovih grafova...

This dissertation examines various properties of commutative rings and modulesusing algebraic combinatorial methods. If the graph is properly associated to a ringR or to an R-module M, then examination of its properties gives useful informationabout the ring R or R-module M.This thesis discusses the determination of the radius of the total graph of acommutative ring R in the case when this graph is connected. Typical extensionssuch as polynomial rings, formal power series, idealization of the R-module M andrelations between the total graph of the ring R and its extensions are also dealtwith.The total graph of a module, a generalization of the total graph of a ring ispresented. Various properties are proved and some relations to the total graph of aring as well as to the zero-divisor graph are established.To gain a better understanding of clean rings and their relatives, the clean graphC¡(R) of a commutative ring with identity is introduced and its various proper-ties established. Further investigation of clean graphs leads to additional resultsconcerning other classes of commutative rings.One of the topics of this thesis is the investigation of the properties of the cor-responding line graph L(T¡(R)) of the total graph T¡(R). The classi¯cation ofall commutative rings whose line graphs of the total graph are planar or toroidalis given. It is shown that for every integer g ¸ 0 there are only ¯nitely manycommutative rings such that °(L(T¡(R))) = g.Also, in this thesis all toroidal graphs which are intersection graphs of idealsof a commutative ring R are classi¯ed. An improvement over the previous resultsconcerning the planarity of these graphs is presented...