Editing Unique factorization domain (section)

== Equivalent conditions for a ring to be a UFD ==
A [[Noetherian ring|Noetherian]] integral domain is a UFD if and only if every [[height (ring theory)|height]] 1 [[prime ideal]] is principal (a proof is given at the end). 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]].

In general, for an integral domain ''A'', the following conditions are equivalent:
# ''A'' is a UFD.
# Every nonzero [[prime ideal]] of ''A'' contains a [[prime element]].{{refn|[[Irving Kaplansky|Kaplansky]]}}
# ''A'' satisfies [[ascending chain condition on principal ideals]] (ACCP), and the [[localization of a ring|localization]] ''S''<sup>&minus;1</sup>''A'' is a UFD, where ''S'' is a [[multiplicatively closed subset]] of ''A'' generated by prime elements. (Nagata criterion)
# ''A'' satisfies [[Ascending chain condition on principal ideals|ACCP]] and every [[irreducible element|irreducible]] is [[prime element|prime]].
# ''A'' is [[atomic domain|atomic]] and every [[irreducible element|irreducible]] is [[prime element|prime]].
# ''A'' is a [[GCD domain]] satisfying [[Ascending chain condition on principal ideals|ACCP]].
# ''A'' is a [[Schreier domain]],<ref>A Schreier domain is an integrally closed integral domain where, whenever ''x'' divides ''yz'', ''x'' can be written as {{nowrap|1=''x'' = ''x''<sub>1</sub> ''x''<sub>2</sub>}} so that ''x''<sub>1</sub> divides ''y'' and ''x''<sub>2</sub> divides ''z''. In particular, a GCD domain is a Schreier domain</ref> and [[atomic domain|atomic]].
# ''A'' is a [[Schreier domain|pre-Schreier domain]] and [[atomic domain|atomic]].
# ''A'' has a [[divisor theory]] in which every divisor is principal.
# ''A'' is a [[Krull domain]] in which every [[divisorial ideal]] is principal (in fact, this is the definition of UFD in Bourbaki.)
# ''A'' is a Krull domain and every prime ideal of height 1 is principal.{{sfnp|Bourbaki|1972|loc=7.3, no 2, Theorem 1}}

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 every prime ideal is generated by a prime element in a PID.

For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains a height one prime ideal (induction on height) that is principal. By (2), the ring is a UFD.