Formally, a unique factorization domain is defined to be an integral domain R in which every nonzero nonunit[?] x of R can be written as a product of irreducible elements of R:
The uniqueness part is sometimes hard to verify, which is why the following equivalent definition is useful: a unique factorization domain is an integral domain R in which every nonzero nonunit can be written as a product of prime elements of R.
All principal ideal domains are UFD's; this includes the integers, all fields, all polynomial rings K[X] where K is a field, and the Gaussian integers Z[i].
In general, if R is a UFD, then so is the polynomial ring R[X]. By induction, we therefore see that the polynomial rings Z[X_{1},...,X_{n}] as well as K[X_{1},...,X_{n}] (K a field) are UFD's.
The formal power series ring K[[X_{1},...,X_{n}]] over a field K is also a unique factorization domain.
In UFD's, every irreducible element is prime (the converse is true in any integral domain).
Any two (or finitely many) 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.
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