unique factorization domain

The fundamental theorem of arithmetic says every integer factors into primes in essentially one way. A unique factorization domain is any ring where this dream still comes true — where 'essentially one way' is made precise enough to survive in settings far stranger than the integers.

An integral domain R is a unique factorization domain if every nonzero nonunit element can be written as a product of irreducible elements, and this factorization is unique up to reordering the factors and replacing them by associates (multiplication by units). Equivalently, R is a UFD when it is an integral domain in which every irreducible element is prime and the ascending chain condition on principal ideals holds — the latter guaranteeing factorizations exist at all.

The hierarchy is Euclidean ⊂ PID ⊂ UFD ⊂ integral domain, with every inclusion strict. A key stability result is that if R is a UFD then so is the polynomial ring R[x] (a consequence of Gauss's lemma), so Z[x] and k[x_1, ..., x_n] are UFDs even though they are not PIDs. The classic failure of unique factorization is Z[√−5], where 6 = 2·3 = (1 + √−5)(1 − √−5) gives two genuinely different irreducible factorizations — this very failure is what motivated Dedekind's unique factorization of ideals.