Two Minimal Prime Ideals in k[X,Y]/<XY>

[Dragonfall]

Homework Statement

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.

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?

 

[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.

 

[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>

 

[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)(y^m + 1) in R[x, y]/(xy)?
How does xⁿ + y^m factor?