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  1. #1
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    Ideal

    Consider the ring Z of integers.

    (a) Prove that the ideal I = (4) is not a maximal ideal.
    Hint. Find an ideal J of Z such that (4) is a subset of J subset of Z but J does not equal (4) and J does not equal Z.

    (b) Prove that the ideal I = (3) is a maximal ideal.
    Hint. Suppose that J is any ideal of Z such that (3) is a subset of J subset of Z. Prove that J = I or J = Z.
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  2. #2
    Senior Member roninpro's Avatar
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    The hints outline exactly what you need to do. Could you explain what trouble you are having?
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  3. #3
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    I know that an ideal I of a ring R is said to be a maximal ideal if I does not equal R and for every ideal J of R,
    if I is a subset of J subset of R then either J = I or J = R.
    Therefore, there is no ideal strictly between I and R.
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  4. #4
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    So if J = (2) would that hold true in the "hint." Also, how do I then link that back to what I want to prove. Would I do a contradiction?
    Last edited by mathgirl1188; November 1st 2010 at 12:20 PM.
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