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Question on maximal ideals in an integral domain

  1. Aug 6, 2011 #1
    In an integral domain, I found that the number of maximal ideals in a Notherian ring containing a particular element is finite. If the condition is dropped that the ring be Notherian, can anything like this be said?
  2. jcsd
  3. Aug 6, 2011 #2


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    If k is a field, then the polynomial ring k[x,y] is a Noetherian integral domain. (regular too) However, the element x is contained in infinitely many maximal ideals; for example:
    • (x, y)
    • (x, y-1)
    • (x, y-2)
    More generally, a maximal ideal of this ring contains x if and only if it is of the form (x, f(y)) for some non-constant polynomial f that is irreducible over k.
  4. Aug 6, 2011 #3
    Ah, yes. I misstepped then. Thanks!
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