In this note a short proof is
given for a theorem due originally to Deckard and to Cashwell and Everett. The
theorem states that every ring of power series over an integral domain R is a unique
factorization domain if and only if every ring of power series over R in a finite set of
indeterminates is a unique factorization domain. The proof is based on a study of the
structure of the multiplicative semigroups of such rings. Much of the novelty and
most of the brevity of this argument may be accounted for by the fact that
Dilworth’s theorem on the decomposition of partially ordered sets is invoked at a
crucial point in the proof.
