MODERN ALGEBRA (MATH 210) PROBLEM SET 3\frac{1}{2}

The following problems appeared on recent quals. They aren’t to be handed in; they are solely for practice. A useful fact you don’t know yet: if k is a finite field, then k[x] has infinitely many maximal ideals. Recall also that a ring is Noetherian if it contains no infinite strictly ascending chains of ideals

I_{1} ⊊ I_{2} ⊂ · · · .

By a remarkable Hallowe’en-related coincidence, one or two of these will re-appear on Problem Set 4.

1. (spring 2002) Let R be a commutative ring with unit.

(i) Let S be a saturated multiplicative set of R. This means that 1 ∈ S, 0 ∉ S, and xy ∈ S if and only if x ∈ S and y ∈ S. Show that R − S is a union of prime ideals. (Hint: If a ∈ R − S consider ideals J with a ∈ J ⊂ R − S.)

(ii) An element a ∈ R is a zero divisor if ab = 0 for some b ≠ 0. Apply (i) to show that the set of zero divisors is a union of prime ideals of R.

2. (fall 2000)

(a) If A is a commutative ring, show that the polynomial ring A[X] contains infinitely many distinct maximal ideals.

(b) Describe all maximal ideals in the ring of formal power series ℤ[[X]].

3. (fall 1999) Suppose V is a commutative ring such that for all a, b ∈ V either a|b or b|a.

(i) Prove that if I and J are two ideals in V then I ⊂ J or J ⊂ I. (ii) Prove that any finitely generated ideal of V is principal, that is, generated by a single

element. (iii) Prove that if V is a Noetherian valuation ring, then there exists an element t ∈ V

such that any proper nonzero ideal of V is (t^{n}) for some whole number n ≥ 1

4. (fall 1998)

(a) Suppose that A is a Noetherian domain. Show that every nonzero noninvertible element of A can be written as a finite product of irreducible elements.

(b) Give an example of a Noetherian integral domain which is not a unique factorization domain.

Date: Hallowe’en 2002.