is a UFD for all integers 1 ≤ n ≤ 22, but not for n = 23.
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 iteration, a polynomial ring in any number of variables over any UFD (and in particular over a field) 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.
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 any field of characteristic not 2. Klein and Nagata showed that the ring R[X1,...,Xn]/Q is a UFD whenever Q is a nonsingular quadratic form in the X's and n is at least 5. When n=4 the ring need not be a UFD. For example, is not a UFD, because the element equals the element so that and are two different factorizations of the same element into irreducibles.
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.:
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 (Samuel 1964, p.35). Similarly the coordinate ringR[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 (Samuel 1964, p.31).
(the set of the integers = 0 mod 3) is not UFD, since in , we have 90 = 3 × 30 = 6 × 15, and all of 3, 30, 6 and 15 are primes in , since they are not the product of two smaller positive integers in .
(the set of the integers = 1 mod 4) is not UFD, since in , we have 693 = 9 × 77 = 21 × 33, and all of 9, 77, 21 and 33 are primes in , since they are not the product of two smaller positive integers in (see Hilbert number).
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.
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 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.
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 height one prime ideal (induction on height) which is principal. By (2), the ring is a UFD.
^Artin, Michael (2011). Algebra. Prentice Hall. p. 360. ISBN 978-0-13-241377-0.
^A Schreier domain is an integrally closed integral domain where, whenever x divides yz, x can be written as x = x1x2 so that x1 divides y and x2 divides z. In particular, a GCD domain is a Schreier domain
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