Vol. 16, No. 2, 1966

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Unique factorization in power series rings and semigroups

Don Deckard and Lincoln Kearney Durst

Vol. 16 (1966), No. 2, 239–242

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.

Mathematical Subject Classification
Primary: 13.15
Secondary: 16.00
Received: 8 May 1964
Published: 1 February 1966
Don Deckard
Lincoln Kearney Durst