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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
Posted:
October 28, 2011
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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:
http://dx.doi.org/10.1090/S0002-9939-2011-11086-4
PII:
S 0002-9939(2011)11086-4
Received by editor(s):
October 11, 2010
Received by editor(s) in revised form:
February 16, 2011
Posted:
October 28, 2011
Communicated by:
Irena Peeva
Article copyright:
© Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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