Unique factorization in power series rings and semigroups
1966; Mathematical Sciences Publishers; Volume: 16; Issue: 2 Linguagem: Inglês
10.2140/pjm.1966.16.239
ISSN1945-5844
Autores Tópico(s)Scheduling and Timetabling Solutions
ResumoIn 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.Suppose R is a ring and that R + is the additive group of R. Suppose / is a nonvoid set well ordered by <.If N is the additive semigroup of the nonnegative integers and M is the weak product (Chevalley [3]) Π?e/tf> let Ri = IW# + For /, g in R z define
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