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.References
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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