Math 480, Spring 2013, Assignment 14

Carefully define the following terms, then give one example and one non-example of each:

1. Maximal ideal.
2. Point.
3. Prime ideal.
4. Irreducible variety.
5. Irreducible component (of some reducible variety).
6. Associated prime (of an ideal).

For each operation on varieties, describe the corresponding operation on ideals, then describe an algorithm that computes this operation (modulo the problem of computing radicals):

1. Union.
2. Intersection.
3. Zariski closure of the difference.

Solve the following problems:

1. Let \(I, J,\) and \(K\) be ideals. Prove that \(\left(I + J\right)K = IK + JK.\) (Hint: each side is, by definition, the smallest ideal containing certain elements. Which elements?)
2. Working in \(\mathbb{C}[x]\), compute the associated primes of \(\left\langle x^3-3x^2+2x\right\rangle\).
3. Working in \(\mathbb{C}[x,y,u,v]\), compute the associated primes of \(\left\langle xu, xv, yu, yv\right\rangle\). Do you think the variety of this ideal has any singular points?