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Prime Ideals

  1. Oct 7, 2009 #1
    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. jcsd
  3. Oct 7, 2009 #2
    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.
     
  4. Oct 7, 2009 #3
    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>
     
  5. Oct 7, 2009 #4
    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
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