UFD

UFD stands for Unique Factorization Domain, a concept in commutative algebra. An integral domain R is a UFD if every nonzero nonunit element can be written as a product of irreducible elements, and this factorization is unique up to the order of the factors and multiplication by units (associates). Equivalently, R is a UFD if every element can be factored into irreducibles and such factorizations are unique up to associates.

Consequences of being a UFD include that irreducible elements are prime; if p is irreducible and p divides ab, then p divides a or p divides b. Classic examples of UFDs include the ring of integers Z and the polynomial ring k[x] in one variable over a field k. More generally, every principal ideal domain (PID) is a UFD, and every Euclidean domain is a UFD, though the converse need not hold.

Examples and non-examples: Z is a UFD, and the Gaussian integers Z[i] are also a UFD. The ring Z[√−5], however, is not a UFD; for instance, 6 factors as 2·3 and as (1+√−5)(1−√−5), giving non-unique factorizations.

Relation to broader context: UFDs form a key class of integral domains with well-behaved factorization. Not every domain is a UFD; many rings of integers in number fields fail to be UFDs. The property is preserved under localization and, in many cases, under polynomial extension in one variable over a field.