Unique Factorization Domain - Equivalent Conditions For A Ring To Be A UFD

Equivalent Conditions For A Ring To Be A UFD

A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal. Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case it is in fact a principal ideal domain.

There are also equivalent conditions for non-noetherian integral domains. Let A be an integral domain. Then the following are equivalent.

1. A is a UFD.
2. Every nonzero prime ideal of A contains a prime element. (Kaplansky)
3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S−1A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion)
4. A satisfies (ACCP) and every irreducible is prime.
5. A is a GCD domain (i.e., any two elements have a greatest common divisor) satisfying (ACCP).
6. A is a Schreier domain, and every nonzero nonunit can be expressed as a finite product of irreducible elements (that is, A is atomic.)
7. A has a divisor theory in which every divisor is principal.
8. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.)
9. A is a Krull domain and every prime ideal of height 1 is principal.

In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since, in a PID, every prime ideal is generated by a prime element.

Let A be a Zariski ring (e.g., a local noetherian ring).If the completion is a UFD, then A is a UFD.