# Commutative Ring Theory/Principal ideal domains

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Definition (principal ideal domain):

A principal ideal domain is an integral domain ${\displaystyle R}$ 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 ${\displaystyle R}$ be a Bézout domain. Then the following are equivalent:

1. ${\displaystyle R}$ is a principal ideal domain
2. ${\displaystyle R}$ is noetherian
3. The principal ideals of ${\displaystyle R}$ satisfy the ascending chain condition
(On the condition of the axiom of dependent choice.)

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

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

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