Chatters has asked whether a unique factorization domain (UFD) R, equipped with an automorphism σ transitive on the height 1 prime ideals of R, is necessarily Dedekind. If R contains an uncountable field, then Chatters observed that the answer is affirm ative. In this note was show:

Theorem. Let R be a local UFD equipped with an automorphism σ which is transitive on the height 1 prime ideals of R. Suppose that σ induces an automorphism of finite order on the residue field κ of R (for example, if κ is a global or finite field), Then R is Dedekind.