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Math Help - Ideals

  1. #1
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    Ideals

    I'm wondering how to go about proving that certain ideals are principal or prime or maximal. I don't have a grasp of the general method if I'm given an ideal in a ring and told to prove it is maximal or prime, or if I'm given a ring and told to prove that it is (or isn't) a PID.

    For example:

    Prove (3,x) is maximal in Z[x]. Is (2,x) maximal? (5,x)?

    Prove that (3) and (x) are prime ideals in Z[x].

    Is Z[x] a PID? I understand why Z is a PID but can't generalize past it.

    If anyone can also show me some more esoteric examples (since these are pretty standard ones), that'd be great. Thanks!
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Here's some useful theorems:

     I \subseteq R is a prime ideal  \iff R/I is an integral domain.

     I \subseteq R is a maximal ideal  \iff R/I is a field.
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  3. #3
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    Thanks, but I know those and need a little more handholding than that. I'm missing being able to do the step where I show Z[x]/3 or Z[x]/x is an integral domain, or that Z[x]/(3,x) is a field.
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  4. #4
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by brisbane View Post
    Thanks, but I know those and need a little more handholding than that. I'm missing being able to do the step where I show Z[x]/3 or Z[x]/x is an integral domain, or that Z[x]/(3,x) is a field.
     \mathbb{Z}[x]/(x) = \mathbb{Z} ,  \mathbb{Z}[x]/(3) = \mathbb{Z}_3[x] , and  \mathbb{Z}[x]/(3,x) = \mathbb{Z}_3
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  5. #5
    Senior Member roninpro's Avatar
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    For the PID part, you need to check if every ideal can be written in the form (p(x)) for some p(x)\in \mathbb{Z}[x]. Maybe you can see if this is possible with the ideals you have. Can you find p(x) such that (p(x))=(3,x)?
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  6. #6
    MHF Contributor chiph588@'s Avatar
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    In a PID, every non zero prime ideal is also maximal (ask if you'd like to see why). For  \mathbb{Z}[x] consider  (x) . What does this theorem tell us?
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