UFDs

A unique factorization domain (UFD) is an integral domain in which every nonzero element that is not a unit 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. An irreducible element is a nonzero nonunit that cannot be factored into a product of two nonunits. In a UFD, irreducibles are also prime, meaning they divide a product only if they divide one of the factors.

Key properties include that the factorization into irreducibles exists and is unique. Consequently, the study of divisors in a UFD reduces to understanding its irreducible elements, often called atoms.

Examples and facts:

- The ring of integers Z is a UFD. Every nonzero nonunit factors uniquely into primes (up to units ±1 and order).

- Polynomial rings over a field, F[x], and more generally polynomial rings in several variables over a field, are UFDs. If F is a field, F[x] is a principal ideal domain and hence a UFD.

- The ring of Gaussian integers Z[i] is a UFD.

- Not all rings are UFDs. A classic counterexample is Z[√-5], where 6 = 2·3 = (1+√-5)(1-√-5) expresses 6 as a product of irreducibles in two essentially different ways, showing failure of unique factorization.

Relationships with other classes:

- Every principal ideal domain (PID) is a UFD, but not every UFD is a PID.

- UFDs generalize the familiar factorization property from integers to a broader class of rings, preserving a notion of "building blocks" for elements.