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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Length functions on integral domains


Authors: David F. Anderson and Paula Pruis
Journal: Proc. Amer. Math. Soc. 113 (1991), 933-937
MSC: Primary 13G05; Secondary 13A05
MathSciNet review: 1057742
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Abstract: Let $ R$ be an integral domain and $ x \in R$ which is a product of irreducible elements. Let $ l(x)$ and $ L(x)$ denote respectively the inf and sup of the lengths of factorizations of $ x$ into a product of irreducible elements. We show that the two limits, $ \bar l(x)$ and $ \bar L(x)$, of $ l({x^n})/n$ and $ L({x^n})/n$, respectively, as $ n$ goes to infinity always exist. Moreover, for any $ 0 \leq \alpha \leq 1 \leq \beta \leq \infty $, there is an integral domain $ R$ and an irreducible $ x \in R$ such that $ \bar l(x) = \alpha $ and $ \overline L (x) = \beta $.


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DOI: http://dx.doi.org/10.1090/S0002-9939-1991-1057742-1
PII: S 0002-9939(1991)1057742-1
Article copyright: © Copyright 1991 American Mathematical Society