nLab unique factorization domain

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Definition

Let RR be a discrete integral domain. We say that an element rRr\in R 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, 00 is never irreducible, because it is a product of two non-units under the form 0=0.00 = 0.0. It is often included in the definition of an irreducible element that it must be non-zero but this is in fact redudant.

An integral domain RR is a unique factorization domain (UFD for short) if every non-zero non-unit has a factorization u=r 1r nu = r_1 \cdots r_n (where n1n \ge 1) as product of irreducibles and this decomposition is unique up to renumbering and rescaling the irreducibles by units.

Put differently: RR is a UFD precisely when the multiplicative monoid of nonzero principal ideals of RR (which is isomorphic to the quotient monoid Can(R)/R ×Can(R)/R^\times, where Can(R)R\{0}Can(R) \coloneqq R \backslash \{0\} denotes the multiplicative subset of cancellative elements in RR and R ×R^\times denotes the group of units in RR) is a commutative monoid freely generated by irreducible principal ideals. It follows that if KK is the field of fractions of RR, then the quotient group K ×/R ×K^\times/R^\times is an abelian group that is freely generated by the set of cosets fR ×f R^\times with ff ranging over irreducible elements. As a side remark, we observe that in this circumstance the exact sequence

0R ×K ×K ×/R ×00 \to R^\times \to K^\times \to K^\times/R^\times \to 0

splits and there is an isomorphism K ×R ×(K ×/R ×)K^\times \cong R^\times \oplus (K^\times/R^\times) of abelian groups.

Examples

Properties

  • 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:

Proposition

Let RR be an integral domain. Then, RR is a UFD iff every non-zero prime ideal contains a non-zero non-unit prime element.

See also

References

Last revised on August 7, 2022 at 23:20:04. See the history of this page for a list of all contributions to it.