Let be an integral domain. Prove that if
- Any two nonzero elements have a greatest common divisor which can be written for some and
- For any sequence such that divides , there exists such that for all there is a unit such that ,
then is a principal ideal domain.
Let be a nonzero ideal. We can consider to be partially ordered by divisibility as follows: say if . Let be a chain; in fact is a sequence with dividing for all . By our hypothesis, there exists such that for all , for some unit . That is, divides for all . Certainly we also have that divides for . Thus is an upper bound of the chain . By Zorn’s Lemma, contains a maximal element with respect to this order; that is, contains elements which are minimal with respect to divisibility.
Now let be two such minimal elements. By our hypothesis, and have a greatest common divisor which we can write . In particular, . Since and are minimal with respect to divisibility, we have that , , and are associates. Thus all divisibility-minimal elements of are associates, and we have (for instance) . Thus is principal, and is a principal ideal domain.