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On a Class of Archimedean Integral Domains

Published online by Cambridge University Press:  20 November 2018

Raymond A. Beauregard  and
David E. Dobbs
Raymond A. Beauregard
Affiliation:
University of Rhode Island, Kingston, Rhode Island
David E. Dobbs
Affiliation:
University of Rhode Island, Kingston, Rhode Island
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Our starting point is an observation in elementary number theory [10, Exercise 26, p. 17]: if a and b are positive integers such that each number in the sequence a, b2, a3, b4, … divides the next, then a = b. Its proof depends only on Z being a unique factorization domain (UFD) whose units are 1, —1. Accordingly, we abstract and say that a (commutative integral) domain R satisfies (*) in case, whenever nonzero elements a and b in R are such that each element in the sequence a, b2, a3, b4, … divides the next, then a and b are associates in R (that is, a = bu for some unit u of R). The main objective of this paper is the study of the class of domains satisfying (*).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 28 , Issue 2 , 01 April 1976 , pp. 365 - 375
Copyright
Copyright © Canadian Mathematical Society 1976

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