Extending ideals from principal ideal domains

Let $D$ be a PID, $E$ a domain containing $D$ as a subring. Is it true that if $d$ is a gcd of $a$ and $b$ in $D$, then $d$ is also a gcd of $a$ and $b$ in $E$?

Solution 1:

In any domain $D$, for $a,b \in D \setminus \{0\}$, if the ideal $\langle a,b \rangle_D = \{xa + yb \ | \ x,y \in D\}$ of $D$ is principal, then any generator $d$ of the ideal is a gcd of $a$ and $b$. (Note that in general gcd's are unique precisely up to units, i.e., the corresponding principal ideal is unique.)

So if $D$ is a PID, there is $d \in D$ such that $\langle a,b \rangle_D = dD$. Now push forward these ideals to $E$:

$\langle a,b \rangle_E = \langle a,b \rangle_D E = (d D) E = dE$.

Thus the ideal of $E$ generated by $a$ and $b$ is still principal and still generated by the same element $d$ (now well-determined up to a unit of $E$; note that the unit group of $E$ could be larger than the unit group of $D$).

So in summary: yes.