Prime and Maximal Ideals

18. Prime and Maximal Ideals Let R be a ring and let I be an ideal of R, where I = R. Consider the quotient ring R/I. Two very natural questions arise: (1) When is R/I a domain? (2) When is R/I a field? Definition-Lemma 18.1. Let R be a ring and let I be an ideal of R. We say that I is prime if whenever ab ∈ I then either a ∈ I or b ∈ I. Then R/I is a domain if and only if I is prime. Proof. Suppose that I is prime. Let x and y be two elements of R/I. Then there are elements a and b of R such that x = a+I and y = b+I. Suppose that xy = 0, but that x = 0, that is, suppose that a ∈ / I. xy = (a + I)(b + I) = ab + I = 0. But then ab ∈ I and as I is prime, b ∈ I. But then y = b + I = I = 0. Thus R/I is an domain. Now suppose that R/I is a domain. Let a and b be two elements of R such that ab ∈ I and suppose that a ∈ / I. Let x = a + I, y = b + I. Then xy = ab + I = 0. As x = 0, and R/I is an domain, y = 0. But then b ∈ I and so I is prime. D Example 18.2. Let R = Z. Then every ideal in R has the form (n) = nZ. It is not hard to see that I is prime iff n is prime. Definition 18.3. Let R be an integral domain and let a be a non-zero element of R. We say that a is prime, if (a) is a prime ideal, not equal to the whole of R. Note that the condition that (a) is not the whole of R is equivalent to requiring that a is not a unit. Definition 18.4. Let R be a ring. Then there is a unique ring homo­ morphism φ : Z −→ R. We say that the characteristic of R is n if the order of the image of φ is finite, equal to n; otherwise the characteristic is 0. Let R be a domain of finite characteristic. Then the characteristic is prime. Proof. Let φ : Z −→ R be a ring homomorphism. Then φ(1) = 1. Note that Z is a cyclic group under addition. Thus there is a unique map that 1 MIT OCW: 18.703 Modern Algebra Prof. James McKernan sends 1 to 1 and is a group homomorphism. Thus φ is certainly unique and it is not hard to check that in fact φ is a ring homomorphism. Now suppose that R is an integral domain. Then the image of φ is an integral domain. In particular the kernel I of φ is a prime ideal. Suppose that I = (p). Then the image of φ is isomorphic to R/I and so the characteristic is equal to p. D Another, obviously equivalent, way to define the characteristic n is to take the minimum non-zero positive integer such that n1 = 0. Example 18.5. The characteristic of Q is zero. Indeed the natural map Z −→ Q is an inclusion. Thus every field that contains Q has characteristic zero. On the other hand Zp is a field of characteristic p. Definition 18.6. Let I be an ideal. We say that I is maximal if for every ideal J, such that I ⊂ J, either J = I or J = R. Proposition 18.7. Let R be a commutative ring. Then R is a field iff the only ideals are {0} and R. Proof. We have already seen that if R is a field, then R contains no non-trivial ideals. Now suppose that R contains no non-trivial ideals and let a ∈ R. Suppose that a = 0 and let I = (a). Then I = {0}. Thus I = R. But then 1 ∈ I and so 1 = ba. Thus a is a unit and as a was arbitrary, R is a field. D Theorem 18.8. Let R be a commutative ring. Then R/M is a field iff M is a maximal ideal. Proof. Note that there is an obvious correspondence between the ideals of R/M and ideals of R that contain M . The result therefore follows immediately from (18.7). D Corollary 18.9. Let R be a commutative ring. Then every maximal ideal is prime. Proof. Clear as every field is an integral domain. D Example 18.10. Let R = Z and let p be a prime. Then I = (p) is not only prime, but it is in fact maximal. Indeed the quotient is Zp . Example 18.11. Let X be a set and let R be a commutative ring and let F be the set of all functions from X to R. Let x ∈ X be a point of X and let I be the ideal of all functions va0nishing at x. Then F/I is isomorphic to R. Thus I is prime iff R is an integral domain and I is maximal iff R is a field. For example, take X = [0, 1] and R = R. In this case it 2 MIT OCW: 18.703 Modern Algebra Prof. James McKernan turns out that every maximal ideal is of the same form (that is, the set of functions vanishing at a point). Example 18.12. Let R be the ring of Gaussian integers and let I be the ideal of all Gaussian integers a + bi where both a and b are divisible by 3. I claim that I is maximal. I will give two ways to prove this. Method I: Suppose that I ⊂ J ⊂ R is an ideal, not equal to I. Then there is an element a + bi ∈ J, where 3 does not divide one of a or b. It follows that 3 does not divide a2 + b2 . But c = a2 + b2 = (a + bi)(a − bi) ∈ J, as a + bi ∈ J and J is an ideal. As 3 does not divide c, we may find integers r and s such that 3r + cs = 1. As c ∈ J, cs ∈ J and as 3 ∈ I ⊂ J, 3r ∈ J as well. But then 1 ∈ J and J = R. Method II: Suppose that (a + bi)(c + di) ∈ I. Then 3|(ac − bd) and 3|(ad + bc). Suppose that a + bi ∈ / I. Adding the two results above we have 3|(a + b)c + (a − b)d. Now either 3 divides a and it does not divide b, or vice-versa, or the same is true, with a + b replacing a and a − b replacing b, as can be seen by an easy case-by-case analysis. Suppose that 3 divides a whilst 3 does not divide b. Then 3|bd and so 3|d as 3 is prime. Similarly 3|c. It follows that c + di ∈ I. Similar analyses pertain in the other cases. Thus I is prime, so that the quotient R/I is an integral domain. As the quotient is finite (easy check) it follows that the quotient is a field, so that I is maximal. It turns out that R/I is a field with nine elements. Now suppose that we replace 3 by 5 and look at the resulting ideal J. I claim that J is not maximal. Indeed consider x = 2 + i and y = 2 − i. Then xy = (2 + i)(2 − i) = 4 + 1 = 5, so that xy ∈ J, whilst neither x nor y are in J, so that J is not even prime. 3 MIT OCW: 18.703 Modern Algebra Prof. James McKernan