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?

Questions:

1. I was wondering about a couple of things. 1.) Would you be willing to briefly recap the notion of the Associated Primes and how they are calculated? I remember you saying there was an algorithm, and the textbook did as well, but for some reason I do not have it in my notes, and 2.) Will you be posting answers for this and last homework set so that we can make sure we have the right answers when we study for the final? --Robert.Moray (talk) 19:43, 15 May 2013 (EDT)