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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$.

References [Enhancements On Off] (What's this?)

  • 1. D. D. Anderson, Some Finiteness Conditions on a Commutative Ring, Houston J. Math., 4, No. 3 (1978), 289-299. MR 514243 (80c:13013)
  • 2. D. D. Anderson and J. L. Mott, Cohen-Kaplansky Domains: Integral Domains with a Finite Number of Irreducible Elements, J. Algebra, 148 (1992), 17-41. MR 1161563 (93e:13041)
  • 3. I.S. Cohen and I. Kaplansky, Rings with a Finite Number of Primes, Trans. Amer. Math. Soc., 60 (1946), 468-477. MR 0019595 (8:434a)
  • 4. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Appl. Math., vol. 90, Kingston, 1992, 513-516. MR 1204267 (93j:13001)

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Additional Information

Jim Coykendall
Affiliation: Department of Mathematics, North Dakota State University, Fargo, North Dakota 58108-6050

Chris Spicer
Affiliation: Department of Mathematical Sciences, Morningside College, Sioux City, Iowa 51106-1717

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.

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