All , hence all Euclidean domains, are UFDs. In particular, the integers (also see Fundamental theorem of arithmetic), the Gaussian integers and the Eisenstein integers are UFDs.

principal ideal domains

If R is a UFD, then so is R[X], the with coefficients in R. Unless R is a field, R[X] is not a principal ideal domain. By induction, a polynomial ring in any number of variables over any UFD (and in particular over a field or over the integers) is a UFD.

ring of polynomials

The ring K[[X₁, ..., X_n]] over a field K (or more generally over a regular UFD such as a PID) is a UFD. On the other hand, the formal power series ring over a UFD need not be a UFD, even if the UFD is local. For example, if R is the localization of k[x, y, z]/(x² + y³ + z⁷) at the prime ideal (x, y, z) then R is a local ring that is a UFD, but the formal power series ring R[[X]] over R is not a UFD.

formal power series

The states that every regular local ring is a UFD.

Auslander–Buchsbaum theorem

${\displaystyle \mathbb {Z} \left[e^{\frac {2\pi i}{n}}\right]}$ is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.

Mori showed that if the completion of a , such as a Noetherian local ring, is a UFD, then the ring is a UFD.^[1] The converse of this is not true: there are Noetherian local rings that are UFDs but whose completions are not. The question of when this happens is rather subtle: for example, for the localization of k[x, y, z]/(x² + y³ + z⁵) at the prime ideal (x, y, z), both the local ring and its completion are UFDs, but in the apparently similar example of the localization of k[x, y, z]/(x² + y³ + z⁷) at the prime ideal (x, y, z) the local ring is a UFD but its completion is not.

Zariski ring

Let ${\displaystyle R}$ be a field of any characteristic other than 2. Klein and Nagata showed that the ring R[X₁, ..., X_n]/Q is a UFD whenever Q is a nonsingular quadratic form in the Xs and n is at least 5. When n = 4, the ring need not be a UFD. For example, R[X, Y, Z, W]/(XY − ZW) is not a UFD, because the element XY equals the element ZW so that XY and ZW are two different factorizations of the same element into irreducibles.

The ring Q[x, y]/(x² + 2y² + 1) is a UFD, but the ring Q(i)[x, y]/(x² + 2y² + 1) is not. On the other hand, The ring Q[x, y]/(x² + y² − 1) is not a UFD, but the ring Q(i)[x, y]/(x² + y² − 1) is. Similarly the coordinate ring R[X, Y, Z]/(X² + Y² + Z² − 1) of the 2-dimensional real sphere is a UFD, but the coordinate ring C[X, Y, Z]/(X² + Y² + Z² − 1) of the complex sphere is not.

[2]

Suppose that the variables X_i are given weights w_i, and F(X₁, ..., X_n) is a of weight w. Then if c is coprime to w and R is a UFD and either every finitely generated projective module over R is free or c is 1 mod w, the ring R[X₁, ..., X_n, Z]/(Z^c − F(X₁, ..., X_n)) is a UFD.^[3]

homogeneous polynomial

In UFDs, every is prime. (In any integral domain, every prime element is irreducible, but the converse does not always hold. For instance, the element z ∈ K[x, y, z]/(z² − xy) is irreducible, but not prime.) Note that this has a partial converse: a domain satisfying the ACCP is a UFD if and only if every irreducible element is prime.

irreducible element

Any two elements of a UFD have a and a least common multiple. Here, a greatest common divisor of a and b is an element d that divides both a and b, and such that every other common divisor of a and b divides d. All greatest common divisors of a and b are associated.

greatest common divisor

Any UFD is . In other words, if R is a UFD with quotient field K, and if an element k in K is a root of a monic polynomial with coefficients in R, then k is an element of R.

integrally closed

Let S be a of a UFD A. Then the localization S⁻¹A is a UFD. A partial converse to this also holds; see below.

multiplicatively closed subset

Parafactorial local ring

Noncommutative unique factorization domain