Editing Unique factorization domain (section)

== Properties ==
Some concepts defined for integers can be generalized to UFDs:
* In UFDs, every [[irreducible element]] is [[prime element|prime]]. (In any integral domain, every prime element is irreducible, but the converse does not always hold.  For instance, the element {{nowrap|''z'' &isin; ''K''[''x'', ''y'', ''z'']/(''z''<sup>2</sup> − ''xy'')}} is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the [[Ascending chain condition on principal ideals|ACCP]] is a UFD if and only if every irreducible element is prime.
* Any two elements of a UFD have a [[greatest common divisor]] and a [[least common multiple]]. Here, a greatest common divisor of ''a'' and ''b'' is an element ''d'' that [[divisor|divides]] both ''a'' and ''b'', and such that every other common divisor of ''a'' and ''b'' divides ''d''. All greatest common divisors of ''a'' and ''b'' are [[associated element|associated]].
* Any UFD is [[integrally closed domain|integrally closed]].  In other words, if ''R'' is a UFD with [[quotient field]] ''K'', and if an element ''k'' in ''K'' is a [[Polynomial root|root]] of a [[monic polynomial]] with [[coefficients]] in ''R'', then ''k'' is an element of ''R''.
* Let ''S'' be a [[multiplicatively closed subset]] of a UFD ''A''. Then the [[localization of a ring|localization]] ''S''<sup>−1</sup>''A'' is a UFD. A partial converse to this also holds; see below.