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).
Read more about this topic: Unique Factorization Domain
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