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 nonprime irreducible elements for every positive integer .

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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.