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Math Help - Greatest Common Denominators - Definition and Uniqueness

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    Super Member Bernhard's Avatar
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    Greatest Common Denominators - Definition and Uniqueness

    On page 274 in Chaper 8 - Euclidean, Principal Ideal and Unique Factorization Domains, Dummit and Foote state Proposition 3 as follows: (see attachment - Proposition 3 ... )

    ================================================== ====================

    Proposition 3

    Let R be an integral domain. If two elements d and d' of R generate the same pricipal ideal, i.e. (d) = (d'), then d' = ud for some unit u in R.

    In particular if d and d' are both greatest common divisors of a and b, then d' = ud for some unit u

    ================================================== =====================

    So we can apply Proposition 3 to \mathbb{Z} yeilding two gcds for each pair of integers a, b.

    For example the gcds of 12 and 18 would be 6 and -6

    So why do D&F make a special definition on page 4 of a unique gcd for  \mathbb{Z}  -  \{0\} by stipulating that the gcd must be positive - see attachment GCD - Properties of the Integers ... for the definition.

    This seems inconsistent .. also why deal with  \mathbb{Z} - \{0\} instead of dealing with \mathbb{Z}. Why do we need a separate definition from Proposition 3. Is it something to do with the fact the the Euclidean Algorithm also gives a positive gcd? I cannot see the motivation for the definition (3) on page 4 of D&F [See attachement GCD - Properties of the Integers ...]

    Can someone with more knowledge please explain why the two definitions are necessary and what the motivation for D&F could be in this matter ... Would appreciate the help.

    Peter
    Last edited by Bernhard; October 13th 2012 at 01:04 AM.
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