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Need help for a maximal ideal quesiton

  1. Jan 24, 2008 #1
    here is the quesiton:

    let I=<2>. prove that I[x] is not a maximal ideal of Z[x] even though I is a maximal ideal of Z

    can someone help me?
     
  2. jcsd
  3. Jan 24, 2008 #2

    morphism

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    What is I[x] supposed to be?
     
  4. Jan 24, 2008 #3

    HallsofIvy

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    Morphism, are you asking this to make sure that wowolala understands it or can anyoe answer?
     
  5. Jan 24, 2008 #4

    morphism

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    Anyone can answer - but who's to tell if what they say is what wowolala had intended. :wink: It's just that I'm not familiar with the notation I[x] when I is an ideal, probably because for me rings have always been unital so that ideals aren't necessarily (sub)rings. But I guess I[x] is the ideal generated by I and {x}?
     
  6. Jan 24, 2008 #5

    HallsofIvy

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    My "guess" would be that, since I is the set of all even integers, I[x] is the set of all polynomials, in the variable x, having even integers for coefficients. That is an ideal of Z[x] but not maximal: there is no single polynomial, having even coefficients, that generates the ideal.
     
  7. Jan 24, 2008 #6

    morphism

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    That's true, but it proves that I[x] isn't principal and not that it isn't maximal.

    I think the fastest way to prove non-maximality is to observe that Z[x]/I[x] isn't a field.
     
  8. Jan 24, 2008 #7

    HallsofIvy

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    Oh, dear, I misread the question!
     
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