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. 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:
rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
Definition Formally, a unique factorization domain is defined to be an integral domain R in which every non-zero element x of R which is not a unit can be written as a finite product of irreducible elements pi of R:
x = p1 p2 ⋅⋅⋅ pn with n ≥ 1 and this representation is unique in the following sense: If q1, ..., qm are irreducible elements of R such that
x = q1 q2 ⋅⋅⋅ qm with m ≥ 1, 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}.
Examples 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.
Z [ e 2 π i / n ] {\displaystyle \textstyle \mathbb {Z} {\bigl [}e^{2\pi i/n}{\bigr ]}} is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23. This is the ring of integers of the cyclotomic field Q ( e 2 π i / n ) {\displaystyle \mathbb {Q} (e^{2\pi i/n})} . For rings of integers of other number fields, see List of number fields with class number one. 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 R {\displaystyle R} 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.
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