Unique Factorization Domain - Examples

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.
  • Any field is trivially a UFD, since every non-zero element is a unit. Examples of fields include rational numbers, real numbers, and complex numbers.
  • If R is a UFD, then so is R, the ring of polynomials with coefficients in R. Unless R is a field, R 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 Auslander–Buchsbaum theorem states that every regular local ring is a UFD.

Further examples of UFDs are:

  • The formal power series ring K] over a field K (or more generally over a PID but not over a UFD).

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