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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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Polynomial extensions of IDF-domains and of IDPF-domains
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by P. Malcolmson and F. Okoh PDF
Proc. Amer. Math. Soc. 137 (2009), 431-437 Request permission

Abstract:

An integral domain is IDF if every non-zero element has only finitely many non-associate irreducible divisors. We investigate when $R$ IDF implies that the ring of polynomials $R[T]$ is IDF. This is true when $R$ is Noetherian and integrally closed, in particular when $R$ is the coordinate ring of a non-singular variety. Some coordinate rings $R$ of singular varieties also give $R[T]$ IDF. Analogous results for the related concept of IDPF are also given. The main result on IDF in this paper states that every countable domain embeds in another countable domain $R$ such that $R$ has no irreducible elements, hence vacuously IDF, and the polynomial ring $R[T]$ is not IDF. This resolves an open question. It is also shown that some subrings $R$ of the ring of Gaussian integers known to be IDPF also have the property that $R[T]$ is not IDPF.
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Additional Information
  • P. Malcolmson
  • Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
  • Email: petem@math.wayne.edu
  • F. Okoh
  • Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
  • Email: okoh@math.wayne.edu
  • Received by editor(s): October 3, 2005
  • Received by editor(s) in revised form: January 18, 2008
  • Published electronically: August 18, 2008
  • Communicated by: Bernd Ulrich
  • © Copyright 2008 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • Journal: Proc. Amer. Math. Soc. 137 (2009), 431-437
  • MSC (2000): Primary 13F20, 13F15; Secondary 13F05, 13B25, 13G05
  • DOI: https://doi.org/10.1090/S0002-9939-08-09531-2
  • MathSciNet review: 2448561