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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.

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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