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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Polynomial extensions of IDF-domains and of IDPF-domains


Authors: P. Malcolmson and F. Okoh
Journal: Proc. Amer. Math. Soc. 137 (2009), 431-437
MSC (2000): Primary 13F20, 13F15; Secondary 13F05, 13B25, 13G05
Published electronically: August 18, 2008
MathSciNet review: 2448561
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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

DOI: http://dx.doi.org/10.1090/S0002-9939-08-09531-2
PII: S 0002-9939(08)09531-2
Keywords: Factorization, polynomials, irreducible, IDF, IDPF
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
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.