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 irreducible elements, uniquely up to order and units.
Unique factorization domains appear in the following chain of class inclusions:
- rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically 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 of a unit u and zero or more irreducible elements pi of R:
- 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,
Most rings familiar from elementary mathematics are UFDs:
- All principal ideal domains, 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.
- If R is a UFD, then so is R[X], the ring of polynomials 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.
- The formal power series ring K[[X1, ..., Xn]] 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.
- The Auslander–Buchsbaum theorem states that every regular local ring is a UFD.
- is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.
- Mori showed that if the completion of a Zariski ring, such as a Noetherian local ring, is a UFD, then the ring is a UFD. 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.
- Let be a field of any characteristic other than 2. Klein and Nagata showed that the ring R[X1, ..., Xn]/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.
- Suppose that the variables Xi are given weights wi, and F(X1, ..., Xn) is a homogeneous polynomial 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, ..., Xn, Z]/(Zc − F(X1, ..., Xn)) is a UFD.
- The quadratic integer ring of all complex numbers of the form , where a and b are integers, is not a UFD because 6 factors as both 2×3 and as . These truly are different factorizations, because the only units in this ring are 1 and −1; thus, none of 2, 3, , and are associate. It is not hard to show that all four factors are irreducible as well, though this may not be obvious. See also Algebraic integer.
- For a square-free positive integer d, the ring of integers of will fail to be a UFD unless d is a Heegner number.
- The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be finite, e.g.:
Some concepts defined for integers can be generalized to UFDs:
- In UFDs, every irreducible element 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.
- Any two elements of a UFD have a greatest common divisor and a least common multiple. Here, a greatest common divisor of a and b is an element d which 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.
- Any UFD is integrally closed. 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.
- Let S be a multiplicatively closed subset of a UFD A. Then the localization S−1A is a UFD. A partial converse to this also holds; see below.
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:
- A is a UFD.
- Every nonzero prime ideal of A contains a prime element.
- 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)
- A satisfies ACCP and every irreducible is prime.
- A is atomic and every irreducible is prime.
- A is a GCD domain satisfying ACCP.
- A is a Schreier domain, and atomic.
- A is a pre-Schreier domain and atomic.
- A has a divisor theory in which every divisor is principal.
- A is a Krull domain in which every divisorial ideal is principal (in fact, this is the definition of UFD in Bourbaki.)
- A is a Krull domain and every prime ideal of height 1 is principal.
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.
- Bourbaki (1972), 7.3, no 6, Proposition 4
- Samuel (1964), p. 35
- Samuel (1964), p. 31
- Artin (2011), p. 360
- 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
- Bourbaki (1972), 7.3, no 2, Theorem 1.
- Artin, Michael (2011). Algebra. Prentice Hall. ISBN 978-0-13-241377-0.
- Bourbaki, N. (1972). Commutative algebra. Paris, Hermann; Reading, Mass., Addison-Wesley Pub. Co. ISBN 9780201006445.
- Hartley, B.; T.O. Hawkes (1970). Rings, modules and linear algebra. Chapman and Hall. ISBN 0-412-09810-5. Chap. 4.
- Lang, Serge (1993), Algebra (Third ed.), Reading, Mass.: Addison-Wesley, ISBN 978-0-201-55540-0, Zbl 0848.13001 Chapter II.5
- David Sharpe (1987). Rings and factorization. Cambridge University Press. ISBN 0-521-33718-6.
- Samuel, Pierre (1964), Murthy, M. Pavman (ed.), Lectures on unique factorization domains, Tata Institute of Fundamental Research Lectures on Mathematics, vol. 30, Bombay: Tata Institute of Fundamental Research, MR 0214579
- Samuel, Pierre (1968). "Unique factorization". The American Mathematical Monthly. 75 (9): 945–952. doi:10.2307/2315529. ISSN 0002-9890. JSTOR 2315529.