中南大学《离散数学》课件-第11章（英文版）.ppt

Copyright2007@byXuDezhi1RingandField

Copyright2007@byXuDezhi2RingDefinition1AringisanalgebraicsystemR,{+,·}satisfyingR,+isanaddtiveAbeliangroupR,·isasemigroup?x,y,z∈R,distributivelawsholdx·(y+z)=x·y+x·z(y+z)·x=y·x+z·x

Copyright2007@byXuDezhi3Sometime,·isomitted.RingRiscommutativeif?x,y∈R,xy=yx.IfRhasamultiplicativeidentity1R≠0,thenwesaythatRisaringwith1.If1R≠0,thenbythefirstdistributivelaw,x0=x(0+0)=x0+x0→x0=0,whilex1R=x

Copyright2007@byXuDezhi4ExamplesofRingsR,{+,×},{0,1}isacommutativeringwith1,whereR=Z,Q,R,CIfRisanadditivegroupZmandifwedefinemultiplicationvia,thenRisacommutativeringwithidentity

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SubringDefinition2AsubringofaringRisasubsetSofRsuchthatthebinaryoperationsonRrestricttobiniaryoperationsonSsothatSisalsoaring.

Copyright2007@byXuDezhi6ZeroDivisorDefinition3Ifx,y∈R－{0}andxy=0,thenxiscalledaleftzerodivisorandyisarightzerodivisor.Thedistinctionbetweenleftandrightdisappears,ofcourse,ifRiscommutative.

ExampleInR=M2(Z),?x,y∈Rsuchthatxy=0,butyx≠0,forinstance,Copyright2007@byXuDezhi71-2-24X=6231Y=

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IntegralDomainandUnitDefinition4AcommutativeringRiscalledanintegraldomainifithasnozerodivisors,orequivalentlyifR–{0}isa(multiplicative)semigroup.

Copyright2007@byXuDezhi9Definition5IfRisaringwith1,thenx∈Risaunitifithasamultiplicativeinverse,i.e.,ifxy=yx=1forsomey∈R.ThesetofunitsinRisdenotedbyU(R).

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DivisionRingandFieldDefinition6GivenaringRwith1RisadivisionringifU(R)=R–{0}RisafieldifRisacommutativedivisionring

Copyright2007@byXuDezhi11

FromRingtoFieldFieldDivision