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:

$R$ is a principal ideal domain
$R$ is noetherian
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 $b \notin ⟨ a ⟩$ . 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 $(a_{n})_{n \in ℕ}$ in $I$ such that $a_{n + 1} \notin ⟨ gcd (a_{1}, \dots, a_{n}) ⟩$ ; indeed, $gcd (a_{1}, \dots, a_{n}) \in I$ since $R$ is a Bézout domain. If we define

b_{n} := gcd (a_{1}, \dots, a_{n})

, we have

⟨ b_{1} ⟩ ⊋ ⟨ b_{2} ⟩ ⊋ \dots

;

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. $◻$