Questions: Prime and Maximal Ideals

Question 1

Which of the following is a prime ideal of Z[x] that is NOT maximal?

• A. (2, x)
• B. (x)
• C. (5, x)
• D. (3, x^2 + 1)

Answer: (B)

The ideal (x) is prime because Z[x]/(x) ≅ Z is an integral domain. But it is not maximal because Z is not a field. By contrast, (2, x) is maximal since Z[x]/(2, x) ≅ Z/2Z ≅ F_2, a field, and similarly for (5, x) and (3, x^2 + 1).

Question 2

In a commutative ring R, the intersection of all prime ideals equals the nilradical (the set of nilpotent elements).

• T. True
• F. False

Answer: True

This is a fundamental result. If a^n = 0, then a^n lies in every prime ideal, and since primes absorb factors, a itself lies in every prime ideal. Conversely, if a is not nilpotent, one constructs a prime ideal avoiding all powers of a using Zorn's lemma applied to the set of ideals disjoint from {1, a, a^2, ...}.

Question 3

In the ring Z/12Z, list all prime ideals.

Model answer: (2)/(12) and (3)/(12), corresponding to the prime factors of 12.

The prime ideals of Z/nZ correspond to the prime ideals of Z containing nZ, which are (p) for primes p dividing n. Since 12 = 2^2 * 3, the prime ideals of Z/12Z are the images of (2) and (3). These are also the maximal ideals, since Z/12Z modulo either is a field.

Question 4

Every commutative ring with unity has at least one maximal ideal.

• T. True
• F. False

Answer: True

This is a direct application of Zorn's lemma. The set of proper ideals of R, ordered by inclusion, is nonempty (it contains (0) if R is not the zero ring) and every chain has an upper bound (the union of a chain of proper ideals is a proper ideal since 1 is not in any member). Zorn's lemma gives a maximal element. Note: the zero ring has no maximal ideal, but by convention the statement requires R ≠ 0.

Question 5

Explain why every maximal ideal is prime, and give an example of a prime ideal that is not maximal.

Model answer: If M is maximal, then R/M is a field, hence an integral domain, so M is prime. The ideal (0) in Z is prime (since Z is a domain) but not maximal (since Z is not a field).

The key is the characterization via quotient rings: M prime iff R/M is a domain, M maximal iff R/M is a field. Every field is a domain, so maximal implies prime. In Z, (0) is prime but (0) subset (2) shows it is not maximal. In k[x, y], (x) is prime (quotient is k[y], a domain) but not maximal (quotient is not a field).