Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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 $.

References [Enhancements On Off] (What's this?)

  • [1] D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra (to appear). MR 1082441 (92b:13028)
  • [2] A. Grams, Atomic domains and the ascending chain condition for principal ideals, Proc. Cambridge Philos. Soc. 75 (1974), 321-329. MR 0340249 (49:5004)
  • [3] D. Rees, Lectures on the asymptotic theory of ideals, London Math. Soc. Lecture Notes Series, No. 113, Cambridge Univ. Press, Cambridge, 1988. MR 988639 (90d:13012)
  • [4] A. Zaks, Half-factorial domains, Israel J. Math. 37 (1980), 281-302. MR 599463 (82b:13009)
  • [5] -, Atomic rings without a.c.c. on principal ideals, J. Algebra 74 (1982), 223-231. MR 644228 (83e:13018)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 13G05, 13A05

Retrieve articles in all journals with MSC: 13G05, 13A05

Additional Information

Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society