# Homework Help: Prime Ideals

1. Oct 7, 2009

### Dragonfall

1. The problem statement, all variables and given/known data
Show that there are exactly two minimal prime ideals in k[X,Y]/<XY>. P is a minimal prime ideal if it is prime and every subset of P that is a prime ideal is actually P. k is a field.

3. The attempt at a solution

Prime ideals of k[X,Y] are <0> and <f> for irreducibles f. But then doesn't every ideal contain <0>? So how can there be other prime ideals?

2. Oct 7, 2009

### aPhilosopher

I think that that definition should be modified to include the word 'non-trivial' somewhere in there. How about:

P is a minimal prime ideal if it is prime and every non-trivial subset of P that is a prime ideal is actually P. k is a field.

3. Oct 7, 2009

### Dragonfall

Alright, but even with that, I'm still not sure how to preceed. It probably has to do with the fact that nontrivial prime ideals of k[X,Y] are generated by irreducible elements. Somehow this translates to two nontrivial minimal prime ideals in k[X,Y]/<XY>

4. Oct 7, 2009

### aPhilosopher

I could be mistaken but I think that the idea of a correspondence between ideals in R and ideals in R/I might called for here.

Edit:

What's (xn + 1)(ym + 1) in R[x, y]/(xy)?
How does xn + ym factor?

Last edited: Oct 7, 2009