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If you have two different common divisors in an integral domain that is not a multiple of each other, is the gcd then equal to the divisor that has the largest norm?

user134489
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  • http://en.wikipedia.org/wiki/Greatest_common_divisor

    I basically try to understand the example given in the section "gcd in commutative rings". What if one of the divisors had larger norm than the other. Would that change something?

    On the other hand, the common divisors would still not be divisible by the other, so I guess this would imply according to the definition of gcd in integral domains, that the gcd still would not exist.

     – user134489 Jan 07 '15 at 18:24
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    Sorry, I had not seen your comment before writing my answer. No it would not change anything that you have a norm. Except you could define a notion of say, "norm-maximal common divisor" if you want. But this would not play well with the usual notion of gcd, which is why it is not done much or at all. – quid Jan 07 '15 at 18:37

2 Answers2

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No, it is not (assuming usual definitions). Indeed, the definition of greatest common divisor is as follows (quoted from Greatest Common Divisor on Wikipedia):

If $R$ is a commutative ring, and $a$ and $b$ are in $R$, then an element $d$ of $R$ is called a common divisor of $a$ and $b$ if it divides both $a$ and $b$ (that is, if there are elements $x$ and $y$ in $R$ such that $dx = a$ and $dy = b$). If $d$ is a common divisor of $a$ and $b$, and every common divisor of $a$ and $b$ divides $d$, then $d$ is called a greatest common divisor of $a$ and $b$.

Thus, in your situation one would say that the elements do not admit a greatest common divisor.

Indeed, there is a specific name for domains where each couple of (non-zero) elements admits a greatest common divisor, these domains are called GCD-domains. Important classes of GCD-domains are UFDs and Bézout domains (i.e., domains where each finitely generated ideal is principal).

(I am a bit puzzled why you mention Gaussian integers, since in fact in the Gaussian integers your situation will never arise, as they are a prinipal ideal domain and thus a GCD-domain.)

quid
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In a domain, gcds, if they exist, are unique only up to associateness, i.e. up to a unit multiple. If we wish the gcd to be unique we must choose a unique representative from each associate equivalence class - sometimes called unit normal representatives (esp. in computational algebra).

For example in $\,\Bbb Z$ we may unit normalize gcds by choosing reps $\ge 0$ and in $\,k[x]\,$ we may choose monic reps. You can find literature about algorithms for such by searching on "unit normal gcd".

See this answer for further discussion of such unit normalization.

Bill Dubuque
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