Transcription of CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and ...
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CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and ...
Page1ofChapter2 CHAPTER 2 RING Basic Definitions and Definitions and CommentsAringRis an abelian group with a multiplicationoperation (a,b) abthat is associative and satisfies the distributive laws:a(b+c)=ab+acand (a+b)c=ab+acfor alla,b,c R. We will always assume thatRhas at least twoelements ,including a multiplicative identity 1 Rsatisfyinga1R=1Ra=afor multiplicative identity is often written simply as 1 ,and the additive identity as 0. Ifa,b ,andcare arbitrary elements ofR ,the following properties are derived quickly from thedefinition of a ring; we sketch the technique in each case.(1)a0=0a=0 [a0+a0=a(0+0)=a0; 0a+0a=(0+0)a=0a](2) ( a)b=a( b)= (ab)[0=0b=(a+( a))b=ab+( a)b ,so ( a)b= (ab); similarly,0=a0=a(b+( b)) =ab+a( b),soa( b)= (ab)](3) ( 1)( 1) = 1 [takea=1,b= 1 in (2)](4) ( a)( b)=ab[replacebby bin (2)](5)a(b c)=ab ac[a(b+( c)) =ab+a( c)=ab+( (ac)) =ab ac](6) (a b)c=ac bc[(a+( b))c=ac+( b)c)=ac (bc)=ac bc](7) 1 = 0 [If 1 = 0 then for allawe havea=a1=a0 = 0 ,soR={0} ,contradicting theassumption thatRhas at least two elements](8) The multiplicati
page 1 of Chapter 2 CHAPTER 2 RING FUNDAMENTALS 2.1 Basic Definitions and Properties 2.1.1 Definitions and Comments A ringRis an abelian group with a multiplication operation (a,b) → abthat is associative and satisfies the distributive laws: a(b+c)=ab+acand (a+ b)c= ab+ acfor all a,b,c∈ R.We will always assume that Rhas at least two elements,including a multiplicative …
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