In mathematics, a unique factorization domain (UFD) (also sometimes called a factorial ring following the terminology of Bourbaki) is a ring in which a statement analogous to the fundamental theorem of arithmetic holds. Specifically, a UFD is an integral domain (a nontrivial commutative ring in which the product of any two non-zero elements is non-zero) in which every non-zero non-unit element can be written as a product of prime elements (or irreducible elements), uniquely up to order and units.

Important examples of UFDs are the integers and polynomial rings in one or more variables with coefficients coming from the integers or from a field.

Unique factorization domains appear in the following chain of class inclusions:

rngsringscommutative ringsintegral domainsintegrally closed domainsGCD domainsunique factorization domainsprincipal ideal domainsEuclidean domainsfieldsalgebraically closed fields


Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R can be written as a product (an empty product if x is a unit) of irreducible elements pi of R and a unit u:

x = u p1 p2 ⋅⋅⋅ pn with n ≥ 0

and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R and w is a unit such that

x = w q1 q2 ⋅⋅⋅ qm with m ≥ 0,

then m = n, and there exists a bijective map φ : {1, ..., n} → {1, ..., m} such that pi is associated to qφ(i) for i ∈ {1, ..., n}.

The uniqueness part is usually hard to verify, which is why the following equivalent definition is useful:

A unique factorization domain is an integral domain R in which every non-zero element can be written as a product of a unit and prime elements of R.


Most rings familiar from elementary mathematics are UFDs:



Some concepts defined for integers can be generalized to UFDs:

Equivalent conditions for a ring to be a UFD

A Noetherian integral domain is a UFD if and only if every height 1 prime ideal is principal (a proof is given at the end). Also, a Dedekind domain is a UFD if and only if its ideal class group is trivial. In this case, it is in fact a principal ideal domain.

In general, for an integral domain A, the following conditions are equivalent:

  1. A is a UFD.
  2. Every nonzero prime ideal of A contains a prime element. (Kaplansky)
  3. A satisfies ascending chain condition on principal ideals (ACCP), and the localization S−1A is a UFD, where S is a multiplicatively closed subset of A generated by prime elements. (Nagata criterion)
  4. A satisfies ACCP and every irreducible is prime.
  5. A is atomic and every irreducible is prime.
  6. A is a GCD domain satisfying ACCP.
  7. A is a Schreier domain,[3] and atomic.
  8. A is a pre-Schreier domain and atomic.
  9. A has a divisor theory in which every divisor is principal.
  10. A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.)
  11. A is a Krull domain and every prime ideal of height 1 is principal.[4]

In practice, (2) and (3) are the most useful conditions to check. For example, it follows immediately from (2) that a PID is a UFD, since every prime ideal is generated by a prime element in a PID.

For another example, consider a Noetherian integral domain in which every height one prime ideal is principal. Since every prime ideal has finite height, it contains a height one prime ideal (induction on height) which is principal. By (2), the ring is a UFD.

See also


  1. ^ Bourbaki, 7.3, no 6, Proposition 4.
  2. ^ Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
  3. ^ A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1 x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain
  4. ^ Bourbaki, 7.3, no 2, Theorem 1.