Commutative Ring Theory/Principal ideal domains

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Definition (principal ideal domain):

A principal ideal domain is an integral domain 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 be a Bézout domain. Then the following are equivalent:

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

Proof: The implication "1. 2." is obvious. Suppose that 3. holds, and let be any ideal. If was non-principal, then whenever , we could find a such that . Hence, starting with an arbitrary and invoking the axiom of dependent choice (applied to a set of finite tuples with an adequate relation) yields a sequence in such that ; indeed, since is a Bézout domain. If we define

, we have ;

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