Definition 1

A Krull domain all of whose divisorial ideals are principal is called a factorial (or unique factorization) domain.

In other words, the group of divisor classes (§ 1, no. 2) is reduced to 0.

Examples (1) Every principal ideal domain is factorial (and, recall, is a Dedekind domain). Conversely, every factorial Dedekind domain is a principal ideal domain by § 2, no. 2, Theorem 1 (c). (2) In particular, if K is a field, the rings K[X] and K[[X]] are factorial domains (see Theorem 2 and Proposition 8 below for generalizations). (3) \* The local ring of a simple point of an algebraic variety is a factorial domain. The ring of germs of functions analytic at the origin of $\mathbf{C}^n$ is a factorial domain. \*