Other meanings of Unique factorization domain
Commutative algebra
A Unique factorization domain is an integral domain in which every nonzero nonunit factors uniquely into irreducible elements, apart from rearrangement and multiplication of factors by units. The concept extends ordinary prime factorization from the integers to broader rings.
The defining property of a unique factorization domain is existence and uniqueness of irreducible factorizations. An integral domain is a commutative ring with identity and no zero divisors; a nonzero nonunit is an element that is neither zero nor invertible. Two elements are associates when they differ by multiplication by a unit. Thus, in a unique factorization domain, the factorizations of an element differ only in the order of factors and replacement of factors by associates.1
For example, in the integers, −60 can be written using the irreducibles 2, 3, and 5, with signs absorbed into a unit. Fields are UFDs vacuously: every nonzero element is a unit, so there are no nonzero nonunits requiring factorization. The definition concerns elements of the ring itself, not arbitrary elements of its fraction field.
In a unique factorization domain, irreducible elements and prime elements coincide. An irreducible cannot be written as a product of two nonunits, while a prime element divides a product only by dividing at least one factor; the implication from irreducible to prime is the substantive consequence of unique factorization.2
Several useful criteria follow. Every finite collection of nonzero elements has a greatest common divisor, determined up to associates, and Bézout identities hold when the relevant ideal is principal. A domain is a UFD precisely when every nonzero nonunit has an irreducible factorization and irreducibles are prime. Every principal ideal domain is therefore a UFD, and every Euclidean domain is a principal ideal domain, giving the familiar chain Euclidean domain → principal ideal domain → unique factorization domain.
Polynomial rings preserve unique factorization under standard hypotheses. Gauss's lemma implies that if R is a UFD, then R[x] is a UFD; repeated application gives R[x1,…,xn] as a UFD for every finite number of variables. Consequently, polynomial rings over a field, such as k[x,y], have unique factorization even though most of them are not principal ideal domains.1
Not every integral domain has the property. In the ring Z[√−5], the equality 6 = 2·3 = (1 + √−5)(1 − √−5) exhibits genuinely different factorizations into irreducibles; this ring is therefore not a UFD. The example shows why being an integral domain alone does not guarantee arithmetic resembling the integers.
Unique factorization is closely tied to divisor theory rather than only to numerical factorization. In a Noetherian normal domain, the UFD condition is equivalent to the triviality of the divisor class group; geometrically, this says that every codimension-one divisor is principal.3
A related subtlety is that failure of unique factorization does not eliminate all useful factorization methods. Rings such as Z[√−5] may have ideals with unique factorization into prime ideals even when elements factor nonuniquely. Localization can also change the answer: inverting selected elements may turn previously nonprincipal behavior into principal behavior, or make a ring a UFD. These distinctions motivate the study of principal ideal domains, Dedekind domains, Krull domains, and divisor class groups as separate levels of arithmetic structure.4
Factorizations are understood up to permutation of factors and replacement of any factor by an associate.
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