abstract algebra - Integral domain with two elements that do not have a gcd

I have the following example of an integral domain with two elements that do not have a gcd from wikipedia:

$R = \mathbb{Z}\left[\sqrt{-3}\,\,\right],\quad a = 4 = 2\cdot 2 = \left(1+\sqrt{-3}\,\,\right)\left(1-\sqrt{-3}\,\,\right),\quad b = \left(1+\sqrt{-3}\,\,\right)\cdot 2.$

The elements $2$ and $1 + \sqrt{−3}$ are two "maximal common divisors" (i.e. any common divisor which is a multiple of $2$ is associated to $2$, the same holds for $1 + \sqrt{−3}$), but they are not associated, so there is no greatest common divisor of a and b.

I can understand it, but how can I prove or strictly explain that $2$ and $1 + \sqrt{−3}$ are two "maximal common divisors" and they are not associated?

Answer

A simple but general way to deduce that this gcd fails to exist is by failure of Euclid's Lemma.

Lemma $\rm\ \ (a,b) = (ac,bc)/c\ \ $ if $\rm\ (ac,bc)\ $ exists $\rm\quad$ [GCD distributive law]

Proof $\rm\quad d\ |\ a,b\iff dc\ |\ ac,bc\iff dc\ |\ (ac,bc)\iff d\mid (ac,bc)/c$

But generally $\rm\ (ac,bc)\ $ need not exist, as is most insightfully viewed as failure of

Euclid's Lemma $\rm\quad a\ |\ bc\ $ and $\rm\ (a,b)=1\ \Rightarrow\ a\ |\ c\ \ $ if $\rm\ (ac,bc)\ $ exists.

Proof $\ \ $ If $\rm\ (ac,bc)\ $ exists then $\rm\ a\ |\ ac,bc\ \Rightarrow\ a\ |\ (ac,bc) = (a,b)\,c = c\ $ by the Lemma.

Hence if $\rm\, a,b,c\, $ fail to satisfy the Euclid Lemma $\Rightarrow,\,$

namely if $\rm\ a\ |\ bc\ $ and $\rm\ (a,b) = 1\ $ but $\rm\ a\nmid c\,$, then one immediately deduces that the gcd $\rm\ (ac,bc)\ $ fails to exist.$\,$ For the special case that $\rm\,a\,$ is an atom (i.e. irreducible), the implication reduces to: atom $\Rightarrow$ prime. So it suffices to find a nonprime atom
in order to exhibit a pair of elements whose gcd fails to exist. This task is a bit simpler, e.g. for $\rm\ \omega = 1 + \sqrt{-3}\ \in\ \mathbb Z[\sqrt{-3}]\ $ we have that the atom $\rm\, 2\ |\ \omega' \omega = 4,\,$ but $\rm\ 2\nmid \omega',\omega,\,$ so $\rm\,2\,$ is not prime. Therefore your gcd $\rm\, (2\omega,\, \omega'\omega) = (2+2\sqrt{-3},\,4)\ $ fails to exist in $\rm\ \mathbb Z[\sqrt{-3}]\,$.

Note that if the gcd $\rm\, (ac,bc)\ $ fails to exist then this implies that the ideal $\rm\ (ac,bc)\ $ is not principal. Therefore we've constructively deduced that the failure of Euclid's lemma immediately yields both a nonexistent gcd and a nonprincipal ideal.

That the $\Rightarrow$ in Euclid's lemma implies that Atoms are Prime $\rm(:= AP)$ is denoted $\rm\ D\ \Rightarrow AP\ $ in the list of domains closely related to GCD domains in my post here. There you will find links to further literature on domains closely related to GCD domains. See especially the referenced comprehensive survey by D.D. Anderson: GCD domains, Gauss' lemma, and contents of polynomials, 2000.

See also my post here for the general universal definitions of $\rm GCD,\, LCM$ and for further remarks on how such $\iff$ definitions enable slick proofs, and see here for another simple example of such.