NonUFDs

NonUFDs, or non-unique factorization domains, are integral domains in which not every nonzero element factors uniquely into irreducibles. In a unique factorization domain (UFD), each nonzero element can be written as a product of irreducibles in essentially one way up to units and order. A non-UFD fails this property by admitting elements with two essentially different factorizations into irreducibles.

A classic example is the ring Z[√-5]. In this ring, 2, 3, and 1±√-5 are irreducible, but the element 6 has two distinct factorizations into irreducibles: 6 = 2·3 = (1+√-5)(1-√-5). Therefore Z[√-5] is not a UFD. In many such rings, irreducibles need not be primes, and divisibility theory diverges from that in the integers.

In algebraic number theory, the occurrence of non-UFDs is tied to the failure of unique factorization of elements despite often having unique factorization of ideals. The obstruction is captured by the class group. A ring of integers is a UFD precisely when its class number is 1. For Q(√-5), the class number is 2, so Z[√-5] is a non-UFD, while ideals still factor uniquely into prime ideals (the Dedekind property).

Other non-UFDs appear in various number fields and related coordinate rings. Studying them often involves comparing factorizations of elements with factorizations of ideals and using invariants such as class numbers and units to understand the extent and nature of the failure of uniqueness.