Unique Factorization Domain - Properties

Properties

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.) Note that this has a partial converse: any Noetherian domain is a UFD if every irreducible element is prime.
  • 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.
  • Any UFD is integrally closed. In other words, if R is an integral domain 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 is a UFD. A partial converse to this also holds; see below.

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