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unique factorization domain

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Definition

Let RR be an 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.

A commutative integral domain RR is a unique factorization domain (UFD for short) if every non-unit has a factorization u=r 1r nu = r_1 \cdots r_n as product of irreducible non-units 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 monoid (R{0})/R ×(R \setminus \{0\})/R^\times, where R ×R^\times denotes the group of units) 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.

Revised on July 2, 2015 21:36:08 by Todd Trimble (67.81.95.215)