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1、September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch01page1Chapter1Domains,ModulesandMatrices1.1Rings,DomainsandFieldsDefinition1.1.1AnonemptysetRiscalledaringifRhastwobinaryoperations,calledadditionandmultiplicationanddenotedbya+bandabrespectively,suchthatforalla,b,c∈Rt
2、hefollowingholds:a+b∈R;(1.1.1)a+b=b+a(thecommutativelaw);(1.1.2)(a+b)+c=a+(b+c)(theassociativelaw);(1.1.3)∃0∈Rsuchthata+0=0+a=a,∀a∈R;(1.1.4)∀a∈R,∃−a∈Rsuchthata+(−a)=0;(1.1.5)ab∈R;(1.1.6)a(bc)=(ab)c(theassociativelaw);(1.1.7)a(b+c)=ab+ac,(b+c)a=ba+ca,(thedistributivelaws).(1.1.8)Rhasanidentity
3、element1ifa1=1aforalla∈R.Riscalledcommutativeifab=ba,foralla,b∈R.(1.1.9)1September21,20158:8Matrices:Algebra,AnalysisandApplications-9inx6inb2108-ch01page22MatricesNotethattheproperties(1.1.2)−(1.1.8)implythata0=0a=0.Ifaandbaretwononzeroelementssuchthatab=0,(1.1.10)thenaandbarecalledzerodivis
4、ors.Definition1.1.2DiscalledanintegraldomainifDisacommutativeringwithoutzerodivisorswhichcontainsanidentityelement1.Theclassicalexampleofanintegraldomainistheringofinte-gersZ.Inthisbookweshallusethefollowingexampleofanintegraldomain.Example1.1.3LetΩ⊂Cnbeanonemptyset.ThenH(Ω)denotestheringoffun
5、ctionsf(z1,...,zn)suchthatforeachζ∈ΩthereexistsanopenneighborhoodO(f,ζ)ofζonwhichfisanalytic.IfΩisopenweassumethatfisdefinedonlyonΩ.IfΩconsistsofonepointζthenHζstandsforH({ζ}).RecallthatΩ⊂Cniscalledconnected,ifintherelativetopologyonΩ,inducedbythestandardtopologyonCn,theonlysubsetsofΩwhichareb
6、othopeninΩandclosedinΩare∅andΩ.NotethatthezeroelementisthezerofunctionofH(Ω)andtheidentityelementistheconstantfunctionwhichisequalto1.ThepropertiesofanalyticfunctionsimplythatH(Ω)isanintegraldomainifandonlyifΩisaconnectedset.Inthisbook,weshallassumethatΩisconnectedunlessotherwisestated.See[Ru
7、d74]and[GuR65]forpropertiesofanalyticfunctionsinoneandseveralcomplexvariables.Definition1.1.4AnonemptyΩ⊂CniscalledadomainifΩisanopenconnectedset.Fora,b∈D,adividesb,(oraisadivisorofb),denotedbya
8、b,ifb=ab1forsomeb1∈D.Anelementaiscalledinvertible
