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The definition of gcd is provided below:

Let R be an principal ideal domain. Let a,b ∈ R. Then there is a greatest common divisor of a and b, that is, an element d that divides both a and b and such that if c divides both a and b, then c divides d. The element d is unique up to the equivalence relation ∼ (a ∼ b if a and b are associate).

Here, a ~ b means a|b and b|a which was proved to be an equivalence relation.

My question is what do they mean by "up to the equivalence relation ~"?

Thanks in advance.

user26857
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Yilin Wang
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4 Answers4

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When you have an equivalence relation you separate the set into equivalence classes. Just as $d$ divides $a$ and $b$, so does any associate of $d$. Any of the associates of $d$ would work as well. As an example, in the Gaussian integers, the units are $1,-1,i,-i$ and the associates of a number are the number multiplied by these. Once you know, for example, that the GCD of $8$ and $3+3i$ is $1+i$, you know that $-1-i, -1+i, 1-i$ are also GCDs and that is all there are. All the elements of the equivalence class are GCDs.

Ross Millikan
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This just means that if $d_1$ is a GCD of $a$ and $b$, and $d_2$ is also a GCD of $a$ and $b$, then $d_1 \sim d_2$.

littleO
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It means that $\,d\,$ and $\,d'$ are gcds $\,\iff d \sim d'.$ The idea is that any generator of the principal ideal generated by $\,a,b\,$ serves as a gcd, and such generators are unique only up to associates. The translation into ideal language is by the following chain of equivalence statements, using universal properties and "contains = divides" for principal ideals.

$$ (a,b) = (d)$$

$$(c) \supseteq (a,b) \iff (c) \supset (d)$$

$$ c\mid a,b \iff c\mid d$$

$$ d \sim \gcd( a,b)$$

Bill Dubuque
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It means that this statement is true as long as a and b are not equivalent.

user247327
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