Exercise A0.V.2.07

Proposition

Let $R$ be a Noetherian domain such that for all non-zero $a,b \in R$ there exist $u, v \in R$ such that $au + bv = \gcd(a,b).$ Then $R$ is a PID.

Proof

Let $I$ be an ideal of $R.$ Since $R$ is Noetherian, $I$ is finitely generated. Let $I = (i_1, \ldots, i_n).$

We prove that $I$ is principal by induction on $n.$

If $n = 1$ then $I$ is principal and we’re done.

Suppose $n \geq 2.$ Let $a, b$ be two distinct generators of $I.$ By our main hypothesis, $I$ contains $u = \gcd(a,b),$ and thus we have the inclusions $\{a,b\} \subseteq (a,b) \subseteq (u) \subseteq I,$ since $(u)$ is the smallest principal ideal containing $(a,b).$ Hence, we can replace the generators $a$ and $b$ with a single generator $u.$ Thus, $I$ is generated by $n - 1$ elements. Using the inductive hypothesis, we conclude that $I$ is principal.

$\blacksquare$