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Math Help - Find maximal ideals

  1. #1
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    Find maximal ideals

    Find all the maximal ideals of :
    (i) Real number IR
    (ii) Integers Z
    (iii) C[x]
    (iv) Z_60

    I got no idea how to do it, so I couldn't attempt these questions. Sorry
    Can some body show me how to do it please?

    Thank you very much
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  2. #2
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    Quote Originally Posted by knguyen2005 View Post
    Find all the maximal ideals of :
    (i) Real number IR
    (ii) Integers Z
    (iii) C[x]
    (iv) Z_60

    I got no idea how to do it, so I couldn't attempt these questions. Sorry
    Can some body show me how to do it please?

    Thank you very much
    Look up your definition for ideals, then try to find ideals in the given ring first. Then you can check which ones are maximal ideals using the definition.
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  3. #3
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    Some results you might find useful:

    1) If R is a principal ideal domain (PID) and I \subseteq R is an ideal then I is maximal iff I is prime.

    2) If R is a ring then <f> is prime iff f is prime
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  4. #4
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    Quote Originally Posted by knguyen2005 View Post
    Find all the maximal ideals of :
    (i) Real number IR

    \mathbb{R} is a field. thus ...

    (ii) Integers Z

    \mathbb{Z} is a PID. the maximal ideals are in the form p\mathbb{Z}, where p is any prime.

    (iii) C[x]

    \mathbb{C}[x] is a PID. the maximal ideals are in the form <x-a>, where a \in \mathbb{C}. that is because \mathbb{C} is algebraically closed.

    (iv) Z_60

    let \{p_1, \cdots , p_k \} be the set of prime divisors of n> 1. then \mathbb{Z}/n\mathbb{Z} has exactly k maximal ideals: m_j = p_j \mathbb{Z}/n \mathbb{Z}, \ 1 \leq j \leq k. (also see (ii))

    I got no idea how to do it, so I couldn't attempt these questions. Sorry
    Can some body show me how to do it please?

    Thank you very much
    see if you can get some ideas now!
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  5. #5
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    Don't worry, I got it now, Thanks again
    Last edited by knguyen2005; October 31st 2009 at 05:02 AM.
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