Анализа прстена и модула придруживањем графова

Abstract

Ova doktorska disertacija prouqava razliqite osobine komutativnih prstena i modula algebarsko kombinatornim metodama. Ako se graf na odgovarajui naqin pridrui prstenu R ili R-modulu M, onda ispitiva em egovih osobina dolazimo do korisnih informacija o R i M. U ovoj tezi odreen je radijus totalnog grafa komutativnog prstena R u sluqaju kada je taj graf povezan. Tipiqna raxirea kao xto su prsten polinoma, prsten formalnih redova, idealizacija R-modula M i prsten matrica Mn(R) takoe su ispitani. Ustanov ene su veze izmeu totalnog grafa polaznog prstena R i totalnih grafova ovih raxirea. Definisaem totalnog grafa modula dato je jedno uopxtee totalnog grafa komutativnog prstena. Ispitane su i dokazane egove razliqite osobine. Ustanov ene su veze sa totalnim grafom prstena kao i neke veze 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 egove osobine. Da im istraivaem qistih grafova dobijeni su dodatni rezultati vezani za druge klase komutativnih prstena. Jedan od predmeta ove teze je i istraivae osobina odgovarajueg linijskog grafa L(T¡(R)) totalnog grafa T¡(R). Data je kompletna klasifikacija svih komutativnih prstena qiji su linijski grafovi totalnog grafa planarni ili toroidalni. Dokazano je da za ceo broj g ¸ 0 postoji samo konaqno mnogo komutativnih prstena takvih da je °(L(T¡(R))) = g. U ovoj tezi su takoe klasifikovani svi toroidalni grafovi koji su grafovi preseka ideala komutativnog prstena R. Dato je i jedno pobo xae postojeih rezultata o planarnosti ovih grafova...

This dissertation examines various properties of commutative rings and modules using algebraic combinatorial methods. If the graph is properly associated to a ring R or to an R-module M, then examination of its properties gives useful information about the ring R or R-module M. This thesis discusses the determination of the radius of the total graph of a commutative ring R in the case when this graph is connected. Typical extensions such as polynomial rings, formal power series, idealization of the R-module M and relations between the total graph of the ring R and its extensions are also dealt with. The total graph of a module, a generalization of the total graph of a ring is presented. Various properties are proved and some relations to the total graph of a ring as well as to the zero-divisor graph are established. To gain a better understanding of clean rings and their relatives, the clean graph C¡(R) of a commutative ring with identity is introduced and its various proper- ties established. Further investigation of clean graphs leads to additional results concerning 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 of all commutative rings whose line graphs of the total graph are planar or toroidal is given. It is shown that for every integer g ¸ 0 there are only ¯nitely many commutative rings such that °(L(T¡(R))) = g. Also, in this thesis all toroidal graphs which are intersection graphs of ideals of a commutative ring R are classi¯ed. An improvement over the previous results concerning the planarity of these graphs is presented...