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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Cohen-Kaplansky domains and the Goldbach conjecture
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by Jim Coykendall and Chris Spicer PDF
Proc. Amer. Math. Soc. 140 (2012), 2227-2233 Request permission

Abstract:

A Cohen-Kaplansky domain is an atomic domain with only a finite number of irreducibles. In this paper, we show that localizations of certain orders of rings of integers are necessarily CK-domains, and then prove there exists a closed form formula for the number of irreducible elements in several different cases of these types of rings. Modulo a variant of the Goldbach Conjecture, this construction allows us to answer a question posed by Cohen and Kaplansky over 60 years ago regarding the construction of a CK-domain containing $n$ nonprime irreducible elements for every positive integer $n$.
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Additional Information
  • Jim Coykendall
  • Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58108-6050
  • Email: jim.coykendall@ndsu.edu
  • Chris Spicer
  • Affiliation: Department of Mathematical Sciences, Morningside College, Sioux City, Iowa 51106-1717
  • Email: spicer@morningside.edu
  • Received by editor(s): October 11, 2010
  • Received by editor(s) in revised form: February 16, 2011
  • Published electronically: October 28, 2011
  • Communicated by: Irena Peeva
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 140 (2012), 2227-2233
  • MSC (2010): Primary 13F15
  • DOI: https://doi.org/10.1090/S0002-9939-2011-11086-4
  • MathSciNet review: 2898686