Let be a discrete integral domain. We say that an element is a unit if it is invertible. A non-unit is called irreducible if it can not be represented as a product of two non-units. Note that then, is never irreducible, because it is a product of two non-units under the form . It is often included in the definition of an irreducible element that it must be non-zero but this is in fact redundant.
An integral domain is a unique factorization domain (UFD for short) if every non-zero non-unit has a factorization (where ) as product of irreducibles and this decomposition is unique up to renumbering and rescaling the irreducibles by units.
Put differently: is a UFD precisely when the multiplicative monoid of nonzero principal ideals of (which is isomorphic to the quotient monoid , where denotes the multiplicative subset of cancellative elements in and denotes the group of units in ) is a commutative monoid freely generated by irreducible principal ideals. It follows that if is the field of fractions of , then the quotient group is an abelian group that is freely generated by the set of cosets with ranging over irreducible elements. As a side remark, we observe that in this circumstance the exact sequence
splits and there is an isomorphism of abelian groups.
The ring of (rational) integers is a UFD.
A principal ideal domain (PID) is a UFD. (In particular, a Euclidean domain is a UFD.) As a partial converse, a Dedekind domain that is a UFD is a PID.
If is a UFD, then its polynomial ring is also a UFD.
If is a UFD, then so is any localization .
A regular local ring (for example, a discrete valuation ring) is a UFD.
If is a UFD and all of its localizations at primes are regular local rings, then the ring of formal power series is a UFD.
For a positive integer, the ring of integers in is a UFD iff is a Heegner number, namely one of the numbers .
If is not integrally closed, then it is not a UFD.
As noted above, a UFD is necessarily integrally closed.
The lattice of principal ideals under the inclusion order is a distributive lattice.
There is a useful characterization of UFDs by Irving Kaplansky:
Let be an integral domain. Then, is a UFD iff every non-zero prime ideal contains a non-zero non-unit prime element.
Henri Lombardi, Claude Quitté (2010): Commutative algebra: Constructive methods (Finite projective modules) Translated by Tania K. Roblo, Springer (2015) (doi:10.1007/978-94-017-9944-7, pdf)
Irving Kaplansky, Commutative rings, rev. ed., the University of Chicago (1974), pdf
John Baez, Hoàng Xuân Sính’s thesis: categorifying group theory (pdf)
Last revised on August 7, 2023 at 13:57:36. See the history of this page for a list of all contributions to it.