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We know that the greatest common divisor (GCD) of a and b is generally denoted gcd$(a, b)$ or $(a,b)$. However in integral domains, the greatest common divisor is often not unique. So what does the notation "$(a,b)$" stand for in this case?

For example, look at the following equation (from here):

the GCD Distributive Law $ (ax,bx)=(a,b)x $

Is "$(a,b)$" an arbitrary gcd of a and b? If so, it is difficult to interpret the equal sign "$=$".

If "$(a,b)$" represents all the gcds of a and b, which means $(a,b)$ is a set, it is hard to interpret the multiplication “$(a,b)x$”.

So I am confused. Could anyone clarify this? Thanks!

Sky subO
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  • it is clear, that GCD has nothing to di with gcd(a,b), so the G stands probably for general , the C maybe for distributive .But what matters is they give you the Distributive Law for a pair of numbers, so you should look in what context you found this. – trula Aug 21 '21 at 19:17
  • An interesting explanation.But according to the post, "GCD" should just mean the greatest common divisor. You can see it here . – Sky subO Aug 21 '21 at 19:57

2 Answers2

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It means that every greatest common divisor of $ax$ and $bx$ can be written as greatest common divisor of $a$ and $b$ times $x$ and also that if you multiply a greatest common divisor of $a$ and $b$ by $x$, then what you get is a greatest common divisor of $ax$ and $bx$.

José Carlos Santos
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My guess would be that $(a,b)$ stands for the ideal generated by the elements $a$ and $b$. So $(a,b)x$ would represent the set $$\{nx: n\in (a,b) \}$$ and equality here "$=$" represents set equality.

In some cases, we can associate a unique value to $(a,b)$. For example, in $\mathbb{Z}$ we have $(2,3)=(1)$ so we say that $1$ is the gcd of $2$ and $3$.

Ryan
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    Let $d$ be a greatest common divisor of $a$ and $b$. It seems that the condition "R is an integral domain" is not sufficient to ensure the equation $(d)=(a)+(b)$. We need other conditions. For example, R is a principal ideal domain. – Sky subO Aug 21 '21 at 20:46