## Polynomial extensions of IDF-domains and of IDPF-domains

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- by P. Malcolmson and F. Okoh
- Proc. Amer. Math. Soc.
**137**(2009), 431-437 - DOI: https://doi.org/10.1090/S0002-9939-08-09531-2
- Published electronically: August 18, 2008
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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.## References

- David F. Anderson and Driss Nour El Abidine,
*Factorization in integral domains. III*, J. Pure Appl. Algebra**135**(1999), no. 2, 107–127. MR**1667552**, DOI 10.1016/S0022-4049(97)00147-3 - D. D. Anderson and Bernadette Mullins,
*Finite factorization domains*, Proc. Amer. Math. Soc.**124**(1996), no. 2, 389–396. MR**1322910**, DOI 10.1090/S0002-9939-96-03284-4 - D. D. Anderson, David F. Anderson, and Muhammad Zafrullah,
*Factorization in integral domains*, J. Pure Appl. Algebra**69**(1990), no. 1, 1–19. MR**1082441**, DOI 10.1016/0022-4049(90)90074-R - D. D. Anderson and M. Zafrullah,
*The Schreier property and Gauss’ lemma*, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)**10**(2007), no. 1, 43–62 (English, with English and Italian summaries). MR**2310957** - George M. Bergman,
*The diamond lemma for ring theory*, Adv. in Math.**29**(1978), no. 2, 178–218. MR**506890**, DOI 10.1016/0001-8708(78)90010-5 - N. Bourbaki,
*Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9*, Springer, Berlin, 2006 (French). Reprint of the 1983 original. MR**2284892** - Jim Coykendall and Muhammad Zafrullah,
*AP-domains and unique factorization*, J. Pure Appl. Algebra**189**(2004), no. 1-3, 27–35. MR**2038561**, DOI 10.1016/j.jpaa.2003.10.036 - Anne Grams and Hoyt Warner,
*Irreducible divisors in domains of finite character*, Duke Math. J.**42**(1975), 271–284. MR**376661** - Robin Hartshorne,
*Algebraic geometry*, Graduate Texts in Mathematics, No. 52, Springer-Verlag, New York-Heidelberg, 1977. MR**0463157** - I. Kaplansky,
*Commutative Rings*, Polygonal Publishing House, Passaic, New Jersey, 1994. - Serge Lang,
*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234** - Peter Malcolmson and Frank Okoh,
*A class of integral domains between factorial domains and IDF-domains*, Houston J. Math.**32**(2006), no. 2, 399–421. MR**2219322**, DOI 10.1017/s0022112068000807 - Moshe Roitman,
*Polynomial extensions of atomic domains*, J. Pure Appl. Algebra**87**(1993), no. 2, 187–199. MR**1224218**, DOI 10.1016/0022-4049(93)90122-A - Wolmer V. Vasconcelos,
*Computational methods in commutative algebra and algebraic geometry*, Algorithms and Computation in Mathematics, vol. 2, Springer-Verlag, Berlin, 1998. With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR**1484973**, DOI 10.1007/978-3-642-58951-5 - M. Zafrullah, Email communication (2005).
- Oscar Zariski and Pierre Samuel,
*Commutative algebra. Vol. II*, The University Series in Higher Mathematics, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London-New York, 1960. MR**0120249**

## Bibliographic 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