Chapter2 - Chapter 2 Ring Fundamentals 2.1 2.1.1 Basic...

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Chapter 2 Ring Fundamentals 2.1 Basic Defnitions and Properties 2.1.1 Defnitions and Comments A ring R is an abelian group with a multiplication operation ( a,b ) ab that is associative and satisfes the distributive laws: a ( b + c )= ab + ac and ( a + b ) c = ab + ac For all a,b,c R . We will always assume that R has at least two elements, including a multiplicative identity 1 R satisFying a 1 R =1 R a = a For all a in R . The multiplicative identity is oFten written simply as 1, and the additive identity as 0. IF , and c are arbitrary elements oF R , the Following properties are derived quickly From the defnition oF a ring; we sketch the technique in each case. (1) a 0=0 a =0[ a 0+ a 0= a (0+0)= a 0; 0 a +0 a =(0+0) a =0 a ] (2) ( a ) b = a ( b ( ab )[ 0 = 0 b =( a +( a )) b = ab a ) b, so ( a ) b = ( ab )] [0 = a a ( b b )) = ab + a ( b ) , so a ( b ( ab )] (3) ( 1)( 1) = 1 [take a ,b = 1 in (2)] (4) ( a )( b ab [replace b by b in (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 6 = 0 [IF 1 = 0 then For all a we have a = a 1= a 0 = 0, so R = { 0 } , contradicting the assumption that R has at least two elements] (8) The multiplicative identity is unique [IF 1 0 is another multiplicative identity then 1=11 0 0 ] 2.1.2 Defnitions and Comments IF a and b are nonzero but ab = 0, we say that a and b are zero divisors ;i F a R and For some b R we have ab = ba = 1, we say that a is a unit or that a is invertible . Note that ab need not equal ba ; iF this holds For all R , we say that R is a commutative ring . 1
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2 CHAPTER 2. RING FUNDAMENTALS An integral domain is a commutative ring with no zero divisors. A division ring or skew feld is a ring in which every nonzero element a has a multi- plicative inverse a 1 (i.e., aa 1 = a 1 a = 1). Thus the nonzero elements form a group under multiplication. A feld is a commutative division ring. Intuitively, in a ring we can do addition, subtraction and multiplication without leaving the set, while in a Feld (or skew Feld) we can do division as well. Any fnite integral domain is a feld . To see this, observe that if a 6 = 0, the map x ax , x R , is injective because R is an integral domain. If R is Fnite, the map is surjective as well, so that ax = 1 for some x . The characteristic of a ring R (written Char R ) is the smallest positive integer such that n 1 = 0, where n 1 is an abbreviation for 1 + 1 + ... 1( n times). If n 1 is never 0, we say that R has characteristic 0 . Note that the characteristic can never be 1, since 1 R 6 =0 . I f R is an integral domain and Char R 6 = 0, then Char R must be a prime number. ±or if Char R = n = rs where r and s are positive integers greater than 1, then ( r 1)( s 1) = n 1 = 0, so either r 1or s 1 is 0, contradicting the minimality of n . A subring of a ring R is a subset S of R that forms a ring under the operations of addition and multiplication deFned on R . In other words, S is an additive subgroup of R that contains 1 R and is closed under multiplication. Note that 1 R is automatically the multiplicative identity of S , since the multiplicative identity is unique (see (8) of 2.1.1).
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This note was uploaded on 11/14/2011 for the course MATH 367 taught by Professor Sdd during the Spring '11 term at Middle East Technical University.

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Chapter2 - Chapter 2 Ring Fundamentals 2.1 2.1.1 Basic...

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