35 proof exercise 2 example 51 the set z under the

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Proof. Exercise. 2 Example 5.1 The set Z under the usual rules of multiplication and addition forms a ring. 2 Example 5.2 For n > 1, the set Z n under the rules of multiplication and addition defined in § 2.3 forms a ring. Note that Z n with n = 1 does not satisfy our definition, since our definition requires that in a ring R , 1 R 6 = 0 R , and in particular, R must contain at least two elements. Actually, if we have an algebraic structure that satisfies all the requirements of a ring except that 1 R = 0 R , then it is easy to see that R consists of the single element 0 R , where 0 R + 0 R = 0 R and 0 R · 0 R = 0 R . 2 Example 5.3 The set Q of rational numbers under the usual rules of multiplication and addition forms a ring. 2 Let R be a ring. The characteristic of R is defined as the exponent of the underlying additive group. Alterna- tively, the characteristic if the least positive integer m such that m · 1 R = 0 R , if such an m exists, and is zero otherwise. For a, b R , we say that b divides a , written b | a , if there exists c R such that a = bc , in which case we say that b is a divisor of a . Note that parts 1-5 of Theorem 1.1 holds for an arbitrary ring. 5.1.1 Units and Fields Let R be a ring. We call u R a unit if it has a multiplicative inverse, i.e., if uu 0 = 1 R for some u 0 R . It is easy to see that the multiplicative inverse of u , if its exists, is unique, and we denote it by u - 1 ; also, for a R , we may write a/u to denote au - 1 . It is clear that a unit u divides every a R . We denote the set of units R * . It is easy to verify that the set R * is closed under multiplication, from which it follows that R * is an abelian group, called the multiplicative group of units of R . If R * contains all non-zero elements of R , then R is called a field . Example 5.4 The only units in the ring Z are ± 1. Hence, Z is not a field. 2 Example 5.5 For n > 1, the units in Z n are the residue classes [ a mod n ] with gcd( a, n ) = 1. In particular, if n is prime, all non-zero residue classes are units, and conversely, if n is composite, some non-zero residue classes are not units. Hence, Z n is a field if and only if n is prime. 2 Example 5.6 Every non-zero element of Q is a unit. Hence, Q is a field. 2 5.1.2 Zero divisors and Integral Domains Let R be a ring. An element a R is called a zero divisor if a 6 = 0 and there exists non-zero b R such that ab = 0 R . If R has no zero divisors, then it is called an integral domain . Put another way, R is an integral domain if and only if ab = 0 R implies a = 0 R or b = 0 R for all a, b R . Note that if u is a unit in R , it cannot be a zero divisor (if ub = 0 R , then multiplying both sides of this equation by u - 1 yields b = 0 R ). In particular, it follows that any field is an integral domain. 36
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Example 5.7 Z is an integral domain. 2 Example 5.8 For n > 1, Z n is an integral domain if and only if n is prime. In particular, if n is composite, so n = n 1 n 2 with 1 < n 1 , n 2 < n , then [ n 1 ] and [ n 2 ] are zero divisors: [ n 1 ][ n 2 ] = [0], but [ n 1 ] 6 = [0] and [ n 2 ] 6 = [0]. 2 Example 5.9 Q is an integral domain. 2 We have the following “cancellation law”: Theorem 5.3 If R is a ring, and
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