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Cohen-Kaplansky domains and the Goldbach conjecture


Authors: Jim Coykendall and Chris Spicer
Journal: Proc. Amer. Math. Soc. 140 (2012), 2227-2233
MSC (2010): Primary 13F15
Published electronically: October 28, 2011
MathSciNet review: 2898686
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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

DOI: https://doi.org/10.1090/S0002-9939-2011-11086-4
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
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.