Commutative Ring Theory/Principal ideal domains

Definition (principal ideal domain):

A principal ideal domain is an integral domain $R$ whose every ideal is principal.

Proposition (a Bézout domain is principal if and only if it is Noetherian or satisfies the ascending chain condition for principal ideals):

Let $R$ be a Bézout domain. Then the following are equivalent:

1. $R$ is a principal ideal domain
2. $R$ is noetherian
3. The principal ideals of $R$ satisfy the ascending chain condition

(On the condition of the axiom of dependent choice.)

Proof: The implication "1. $\Rightarrow$ 2." is obvious. Suppose that 3. holds, and let $I\leq R$ be any ideal. If $I$ was non-principal, then whenever $a\in I$, we could find a $b\in I$ such that $b\notin \langle a\rangle$. Hence, starting with an arbitrary $a_{1}\in I$ and invoking the axiom of dependent choice (applied to a set of finite tuples with an adequate relation) yields a sequence $(a_{n})_{n\in \mathbb {N} }$ in $I$ such that $a_{n+1}\notin \langle \gcd(a_{1},\ldots ,a_{n})\rangle$; indeed, $\gcd(a_{1},\ldots ,a_{n})\in I$ since $R$ is a Bézout domain. If we define

$b_{n}:=\gcd(a_{1},\ldots ,a_{n})$, we have $\langle b_{1}\rangle \supsetneq \langle b_{2}\rangle \supsetneq \cdots$;

thus, we have defined an ascending chain of principal ideals of $R$ that does not stabilize. Finally, every principal ideal domain must be noetherian, since being noetherian is equivalent to all ideals being finitely generated. $\Box$