Abstract Algebra Theory and Applications - Computer Science ...

244 CHAPTER 14 RINGSTheorem 14.15 Let R be a commutative ring with identity and M an idealin R. Then M is a maximal ideal of R if and only if R/M is a field.Proof. Let M be a maximal ideal in R. If R is a commutative ring, thenR/M must also be a commutative ring. Clearly, 1 + M acts as an identityfor R/M. We must also show that every nonzero element in R/M has aninverse. If a + M is a nonzero element in R/M, then a /∈ M. Define I to bethe set {ra + m : r ∈ R and m ∈ M}. We will show that I is an ideal in R.The set I is nonempty since 0a + 0 = 0 is in I. If r 1 a + m 1 and r 2 a + m 2are two elements in I, then(r 1 a + m 1 ) − (r 2 a + m 2 ) = (r 1 − r 2 )a + (m 1 − m 2 )is in I. Also, for any r ∈ R it is true that rI ⊂ I; hence, I is closedunder multiplication and satisfies the necessary conditions to be an ideal.Therefore, by Proposition 14.2 and the definition of an ideal, I is an idealproperly containing M. Since M is a maximal ideal, I = R; consequently,by the definition of I there must be an m in M and a b in R such that1 = ab + m. Therefore,1 + M = ab + M = ba + M = (a + M)(b + M).Conversely, suppose that M is an ideal and R/M is a field. Since R/Mis a field, it must contain at least two elements: 0 + M = M and 1 + M.Hence, M is a proper ideal of R. Let I be any ideal properly containing M.We need to show that I = R. Choose a in I but not in M. Since a + M is anonzero element in a field, there exists an element b + M in R/M such that(a + M)(b + M) = ab + M = 1 + M. Consequently, there exists an elementm ∈ M such that ab + m = 1 and 1 is in I. Therefore, r1 = r ∈ I for allr ∈ R. Consequently, I = R.□Example 18. Let pZ be an ideal in Z, where p is prime. Then pZ is amaximal ideal since Z/pZ ∼ = Z p is a field.An ideal P in a commutative ring R is called a prime ideal if wheneverab ∈ P , then either a ∈ P or b ∈ P .Example 19. It is easy to check that the set P = {0, 2, 4, 6, 8, 10} is anideal in Z 12 . This ideal is prime. In fact, it is a maximal ideal. Proposition 14.16 Let R be a commutative ring with identity. Then P isa prime ideal in R if and only if R/P is an integral domain.