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

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 IDF implies that the ring of polynomials is IDF. This is true when is Noetherian and integrally closed, in particular when is the coordinate ring of a non-singular variety. Some coordinate rings of singular varieties also give 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 such that has no irreducible elements, hence vacuously IDF, and the polynomial ring is not IDF. This resolves an open question. It is also shown that some subrings of the ring of Gaussian integers known to be IDPF also have the property that is not IDPF.

**1.**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**, 10.1016/S0022-4049(97)00147-3**2.**D. D. Anderson and Bernadette Mullins,*Finite factorization domains*, Proc. Amer. Math. Soc.**124**(1996), no. 2, 389–396. MR**1322910**, 10.1090/S0002-9939-96-03284-4**3.**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**, 10.1016/0022-4049(90)90074-R**4.**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****5.**George M. Bergman,*The diamond lemma for ring theory*, Adv. in Math.**29**(1978), no. 2, 178–218. MR**506890**, 10.1016/0001-8708(78)90010-5**6.**N. Bourbaki,*Éléments de mathématique. Algèbre commutative. Chapitres 8 et 9*, Springer, Berlin, 2006 (French). Reprint of the 1983 original. MR**2284892****7.**Jim Coykendall and Muhammad Zafrullah,*AP-domains and unique factorization*, J. Pure Appl. Algebra**189**(2004), no. 1-3, 27–35. MR**2038561**, 10.1016/j.jpaa.2003.10.036**8.**Anne Grams and Hoyt Warner,*Irreducible divisors in domains of finite character*, Duke Math. J.**42**(1975), 271–284. MR**0376661****9.**Robin Hartshorne,*Algebraic geometry*, Springer-Verlag, New York-Heidelberg, 1977. Graduate Texts in Mathematics, No. 52. MR**0463157****10.**I. Kaplansky,*Commutative Rings*, Polygonal Publishing House, Passaic, New Jersey, 1994.**11.**Serge Lang,*Algebra*, Addison-Wesley Publishing Co., Inc., Reading, Mass., 1965. MR**0197234****12.**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****13.**Moshe Roitman,*Polynomial extensions of atomic domains*, J. Pure Appl. Algebra**87**(1993), no. 2, 187–199. MR**1224218**, 10.1016/0022-4049(93)90122-A**14.**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****15.**M. Zafrullah, Email communication (2005).**16.**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**

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:
http://dx.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.