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Math Help - Irreducible in PID

  1. #1
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    Irreducible in PID

    Let D be a princple ideal domain and let p \in D. Prove that <p> is a maximal ideal in D iff p is irreducible.

    Proof.

    Assume p is irreducible, and let p = ab, then either
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  2. #2
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    Quote Originally Posted by tttcomrader View Post
    Let D be a princple ideal domain and let p \in D. Prove that <p> is a maximal ideal in D iff p is irreducible.
    (Actually we should also specify that p is not a unit also).
    Here is half of the argument:
    Say that \left< p \right> is a maximal ideal and p=ab. Then \left< p \right> \subseteq \left< a \right> . Thus either \left< p \right> \subset \left< a \right> or \left< p \right> = \left< a \right>. If the first case then p and a are associate elements so b is a unit. If the second case then \left< a \right> = D (because \left< p \right> is maximal) and thus a is a unit.
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  3. #3
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    Conversely, assume p = ab, with b a unit.

    Suppose <p> \subseteq J \subseteq D, I need to show that J = D.

    Can I say that since <p> = <a> for any element a in D, then p must contain all the elements in D, thus J = D?
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