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Math Help - Principal Ideals of Rationals

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    Principal Ideals of Rationals

    What is the definition of a Principal Ideal in a integral domain? is it the same thing is a Principal Ideal in a Ring? I suspect they are different.
    Last edited by fuzbyone; December 7th 2009 at 05:15 AM.
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    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by fuzbyone View Post
    So, I know that an ideal of the rational numbers must include all denominator values:

    a/b in I = Ideal
    b/d in Q = rationals, for any integer d

    (a/b)(b/d) = (a/d) in I

    Therefore the denominator of I can be any integer d.

    Near the end the numerator must be an ideal of Z = set of all integers. Some element from nZ, a multiple of n.

    OK, I get stuck when I try to show it's principal: the trouble is when I need to show that there is a element from this ideal that generates the entire set I. I don't understand how you can express the generator of the principal Ideal with the lowest term, because the lowest term has a infinite denominator. If the denominator is infinite then it doesn't exactly exist. If a smallest positive element doesn't exist then there cannot exist a principal ideal of the rationals.
    Firstly, note that \mathbb{Q} is a field.

    Secondly, let R be any ring, I an ideal of R. Then what happens if 1 \in I?

    Now, apply these two things to your problem.
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    Quote Originally Posted by Swlabr View Post
    Firstly, note that \mathbb{Q} is a field.

    Secondly, let R be any ring, I an ideal of R. Then what happens if 1 \in I?

    Now, apply these two things to your problem.
    Thanks for the reply, I can see that anything with the Unit becomes the entire field.

    A new question: What is the definition of a Principal Ideal in a integral domain? is it the same thing as a Principal Ideal in a Ring? I suspect they are different definitions.
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  5. #5
    MHF Contributor Swlabr's Avatar
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    Quote Originally Posted by fuzbyone View Post
    Thanks for the reply, I can see that anything with the Unit becomes the entire field.

    A new question: What is the definition of a Principal Ideal in a integral domain? is it the same thing as a Principal Ideal in a Ring? I suspect they are different definitions.
    What definition do you have for in a ring?
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