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Maximal ideals in Z[X]

  1. Sep 26, 2011 #1
    Since 2Z is a maximal ideal of Z, 2Z[X] is an ideal of Z[X] but it is not maximal since Z[X]/2Z[X]~(Z/2Z)[X] is not a field.

    I'm wondering if a is a maximal ideal of A, when can you say that a[X] is a maximal ideal of A[X]?

    I suppose that for any A which does not have the zero ideal as a maximal ideal, the polynomial X would not be a unit in (A/a)[X]. So... a[X] is a maximal ideal of A[X] if and only if a=0 is maximal in A?
  2. jcsd
  3. Sep 26, 2011 #2
    Almost never, considering that if I is an ideal of A, then I+XA[X] is an ideal that contains I. So For example, 2Z[X] isn't maximal since 2Z+XZ[X] is an ideal that contains it (and that is maximal.

    Here is a picture of all the prime ideals of Z[X]:


    The regular dots represent the maximal ideals. The ugly things like [(2)] or [(0)] are just prime ideals. See http://www.neverendingbooks.org/index.php/grothendiecks-functor-of-points.html [Broken] for more information.
    Last edited by a moderator: May 5, 2017
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