2 Attachment(s)

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 ... )

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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

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So we can apply Proposition 3 to $\displaystyle \mathbb{Z}$ 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 $\displaystyle \mathbb{Z} - \{0\}$ 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 $\displaystyle \mathbb{Z} - \{0\}$ instead of dealing with $\displaystyle \mathbb{Z}$. 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