Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9939-08-09531-2
Published electronically: August 18, 2008
MathSciNet review: 2448561
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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

  • 1. D. F. Anderson and D. N. El Abidine, Factorization in integral domains. III, J. Pure Appl. Algebra 135 (1999), 107-127. MR 1667552 (2000k:13020)
  • 2. D. D. Anderson and B. Mullins, Finite factorization domains, Proc. Amer. Math. Soc. 124 (1996), 389-396. MR 1322910 (96i:13001)
  • 3. D. D. Anderson, D. F. Anderson, and M. Zafrullah, Factorization in integral domains, J. Pure Appl. Algebra 69 (1990), 1-19. MR 1082441 (92b:13028)
  • 4. D. D. Anderson and M. Zafrullah, The Schreier property and Gauss's lemma, Boll. Unione Mat. Itali. Sez. B Artic. Ric. Mat. (8) 10 (2007), 43-62. MR 2310957 (2008c:13001)
  • 5. G. M. Bergman, The diamond lemma for ring theory, Adv. Math. 29 (1978), 178-218. MR 506890 (81b:16001)
  • 6. N. Bourbaki, Commutative algebra, Elements of Mathematics, Chapters 8 and 9, Springer-Verlag, 1989. MR 2284892 (2007h:13001)
  • 7. J. Coykendall and M. Zafrullah, AP-domains and unique factorization, J. Pure Appl. Algebra 189 (2004), 27-35. MR 2038561 (2004j:13029)
  • 8. A. Grams and H. Warner, Irreducible divisors in domains of finite character, Duke Math. J. 42 (1975), 271-284. MR 0376661 (51:12836)
  • 9. R. Hartshorne, Algebraic Geometry, Springer-Verlag, 1977. MR 0463157 (57:3116)
  • 10. I. Kaplansky, Commutative Rings, Polygonal Publishing House, Passaic, New Jersey, 1994.
  • 11. S. Lang, Algebra, Third Edition, Addison Wesley, 1993. MR 0197234 (33:5416)
  • 12. P. Malcolmson and F. Okoh, A class of integral domains between factorial domains and IDF-domains, Houston J. Math. 32 (2006), no. 2, 399-421. MR 2219322 (2007b:13034)
  • 13. M. Roitman, Polynomial extensions of atomic domains, J. Pure Appl. Algebra 87 (1993), 187-199. MR 1224218 (94e:13042)
  • 14. W. V. Vasconcelos, Computational methods in commutative algebra and algebraic geometry, Springer, 1998. MR 1484973 (99c:13048)
  • 15. M. Zafrullah, Email communication (2005).
  • 16. O. Zariski and P. Samuel, Commutative Algebra, Von Nostrand, Princeton, 1960. MR 0120249 (22:11006)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13F20, 13F15, 13F05, 13B25, 13G05

Retrieve articles in all journals with MSC (2000): 13F20, 13F15, 13F05, 13B25, 13G05


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: https://doi.org/10.1090/S0002-9939-08-09531-2
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.

American Mathematical Society