Definition of gcd's in Euclidean domains

Question

In a course, we defined $\gcd(a,b)$ in a Euclidean domain to be a common divisor of $a,b$ with greatest possible norm/valuation.

Looking at a (commutative) ring $R$ as a category with $r\rightarrow s\iff r\mid s$, we can define $\gcd(a,b)$ to be the product of $a$ and $b$. I like this definition a lot, but I'm not sure how it generalizes coincides with the previous one, since we didn't ask the valuation $\nu$ to satisfy $a\mid b\implies \nu(a)\leq \nu (b)$.

How to resolve this?

Clarification: I'm not asking for help in unwrapping the categorical definition, which simply says $c\mid a,b\iff c\mid \gcd(a,b)$. I am asking why these two definitions are equivalent in a Euclidean domain, if they are. As I recall, a valuation is not part of the data of a Euclidean domain, only its existence.

Answer 1

The issue here is what conditions one requires on a valuation function. From https://en.wikipedia.org/wiki/Euclidean_domain:

Let $R$ be an integral domain. A Euclidean function [or valuation] on $R$ is a function $f$ from $R \setminus \{0\}$ to the non-negative integers satisfying the following fundamental division-with-remainder property:

• $(EF_1)$ If $a$ and $b$ are in $R$ and $b$ is nonzero, then there are $q$ and $r$ in $R$ such that $a = bq + r$ and either $r = 0$ or $f(r) < f(b)$.

A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. It is important to note that a particular Euclidean function $f$ is not part of the structure of a Euclidean domain: in general, a Euclidean domain will admit many different Euclidean functions.

Most algebra texts require a Euclidean function to have the following additional property:

• $(EF_2)$ For all nonzero $a$ and $b$ in $R$, $f(a) ≤ f(ab)$.

Let me interrupt the Wikipedia quotation here to point out that property $EF_2$ is essentially what you are asking about. If $EF_2$ holds for some valuation $\nu$, then of course $a \mid b \implies \nu(a) \le \nu(b)$.

So one way to paraphrase your question is: What if we are using a definition of valuation that requires $EF_1$ but not $EF_2$?

Now we return to our quote from Wikipedia:

However, one can show that $EF_2$ is superfluous in the following sense: any domain $R$ which can be endowed with a function $g$ satisfying $EF_1$ can also be endowed with a function $f$ satisfying $EF_1$ and $EF_2$: indeed, for $a \in R \setminus \{0\}$ one can define $f(a)$ as follows:

$$f(a) = \min_{x \in R \setminus \{0\}} g(xa)$$

In words, one may define $f(a)$ to be the minimum value attained by $g$ on the set of all non-zero elements of the principal ideal generated by $a$.

So one way to answer your question is: Let's suppose you are working with a valuation $\nu$ that does not satisfy $EF_2$. By the result above, you can switch to another one that does. With respect to that new valuation, the categorical definition of GCD coincides with the "old" definition.

Answer 2

Actually, this question got me thinking, and in another question I gave the following answer : the definition of a $gcd$ as a common divisor with maximal valuation is equivalent to the usual one if and only if the valuation satisfies for all $a,b\in R$:

$a|b\implies \nu(a)\leqslant \nu(b)$ with equality if and only if $a$ and $b$ are associated.

So not only is this definition dependant on the choice of valuation, I am not even completely convinced that you can always choose such a valuation. If someone has an argument...

See here for my complete answer.