Abstract Algebra Theory and Applications - Computer Science ...

238 CHAPTER 14 RINGSProposition 14.3 (Cancellation Law) Let D be a commutative ring withidentity. Then D is an integral domain if and only if for all nonzero elementsa ∈ D with ab = ac, we have b = c.Proof. Let D be an integral domain. Then D has no zero divisors. Letab = ac with a ≠ 0. Then a(b − c) = 0. Hence, b − c = 0 and b = c.Conversely, let us suppose that cancellation is possible in D. That is,suppose that ab = ac implies b = c. Let ab = 0. If a ≠ 0, then ab = a0 orb = 0. Therefore, a cannot be a zero divisor.□The following surprising theorem is due to Wedderburn.Theorem 14.4 Every finite integral domain is a field.Proof. Let D be a finite integral domain and D ∗ be the set of nonzeroelements of D. We must show that every element in D ∗ has an inverse. Foreach a ∈ D ∗ we can define a map λ a : D ∗ → D ∗ by λ a (d) = ad. This mapmakes sense, because if a ≠ 0 and d ≠ 0, then ad ≠ 0. The map λ a isone-to-one, since for d 1 , d 2 ∈ D ∗ ,ad 1 = λ a (d 1 ) = λ a (d 2 ) = ad 2implies d 1 = d 2 by left cancellation. Since D ∗ is a finite set, the map λ amust also be onto; hence, for some d ∈ D ∗ , λ a (d) = ad = 1. Therefore, ahas a left inverse. Since D is commutative, d must also be a right inversefor a. Consequently, D is a field.□For any nonnegative integer n and any element r in a ring R we writer + · · · + r (n times) as nr. We define the characteristic of a ring R to bethe least positive integer n such that nr = 0 for all r ∈ R. If no such integerexists, then the characteristic of R is defined to be 0.Example 12. For every prime p, Z p is a field of characteristic p. ByProposition 2.1, every nonzero element in Z p has an inverse; hence, Z p is afield. If a is any nonzero element in the field, then pa = 0, since the orderof any nonzero element in the abelian group Z p is p.Theorem 14.5 The characteristic of an integral domain is either primeor zero.Proof. Let D be an integral domain and suppose that the characteristicof D is n with n ≠ 0. If n is not prime, then n = ab, where 1 < a < n and