Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
|
   
Available in electronic format
Available in print format 		Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(e) ISSN 0002-9939(p)

     

Gaussian polynomials and invertibility

Author(s): Thomas G. Lucas
Journal: Proc. Amer. Math. Soc. 133 (2005), 1881-1886.
MSC (2000): Primary 13A15, 13B25
Posted: February 24, 2005
MathSciNet review: 2137851
Retrieve article in: PDF
This article is available free of charge

Abstract | References | Similar articles | Additional information

Abstract: The content of a polynomial $f$ over a commutative ring $R$ is the ideal $c(f)$ of $R$ generated by the coefficients of $f$. If $c(fg)=c(f)c(g)$ for each polynomial $g\in R[x]$, then $f$ is said to be Gaussian. If $c(f)$ is an invertible ideal of $R$, then $f$ is Gaussian. An open question has been whether the converse holds for a polynomial whose content is a regular ideal of $R$. The main theorem shows slightly more than this; namely, if $c(f)$ has no nonzero annihilators, then $c(f)Hom_R(c(f),R)=R$.

References:

[A]
D.D. Anderson, Some remarks on the ring $R(X)$, Comment. Math. Univ. St. Pauli 26 (1977), 137-140. MR 0476720 (57:16279)

[AAM]
D.D. Anderson, D.F. Anderson and R. Markanda, The rings $R(X)$ and $R\langle X\rangle$, J. Algebra 95 (1985), 96-115. MR 0797658 (86k:13020)

[AK]
D.D. Anderson and B.G. Kang, Content formulas for polynomials and power series and complete integral closure, J. Algebra 181 (1996), 82-94. MR 1382027 (97c:13014)

[CG]
A. Corso and S. Glaz, Gaussian ideals and the Dedekind-Mertens lemma, in Geometric and combinatorial aspects of commutative algebra (Messina, 1999), 131-143, Lecture Notes in Pure and Appl. Math., 212, Dekker, New York, 2001. MR 1824223 (2002b:13015)

[ES]
P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439-454. MR 0419428 (54:7449)

[G]
R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics, 90, Queen's University Press, Kingston, 1992. MR 1204267 (93j:13001)

[GH]
R. Gilmer and J. Hoffman, A characterization of Prüfer domains in terms of polynomials, Pacific J. Math. 60 (1975), 81-85. MR 0412175 (54:302)

[GV]
S. Glaz and W. Vasconcelos, The content of Gaussian polynomials, J. Algebra 202 (1998), 1-9. MR 1614237 (99c:13003)

[HH1]
W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739-745.MR 1401742 (97e:13015)

[HH2]
W. Heinzer and C. Huneke, The Dedekind-Mertens lemma and the contents of polynomials, Proc. Amer. Math. Soc. 126 (1998), 1305-1309. MR 1425124 (98j:13003)

[H]
J. Huckaba, Commutative rings with zero divisors, Dekker, New York, 1988. MR 0938741 (89e:13001)

[La]
J. Lambek, Lectures on rings and modules, Chelsea, New York, 1986. MR 0419493 (54:7514)

[Li]
J. Lipman, Stable ideals and Arf rings, Amer. J. Math. 93 (1971), 649-685. MR 0282969 (44:203)

[LR]
K.A. Loper and M. Roitman, The content of a Gaussian polynomial is invertible, Proc. Amer. Math. Soc. 133 (2005), 1267-1271.

[Lu1]
T. Lucas, Some results on Prüfer rings, Pacific J. Math. 124 (1986), 333-343. MR 0856167 (87j:13023)

[Lu2]
T. Lucas, Strong Prüfer rings and the ring of finite fractions, J. Pure Appl. Algebra 84 (1993), 59-71.MR 1195418 (93k:13026)

[T1]
H. Tsang, Gauss' lemma, dissertation, University of Chicago, Chicago, 1965.

[T2]
H.T. Tang, Gauss' lemma, Proc. Amer. Math. Soc. 35 (1972), 372-376. MR 0302638 (46:1782)

Similar Articles:

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 13A15, 13B25

Retrieve articles in all Journals with MSC (2000): 13A15, 13B25

Additional Information:

Thomas G. Lucas
Affiliation: Department of Mathematics and Statistics, University of North Carolina Charlotte, Charlotte, North Carolina 28223
Email: tglucas@uncc.edu

DOI: 10.1090/S0002-9939-05-07977-3
PII: S 0002-9939(05)07977-3
Keywords: Gaussian polynomial, content, $Q_0$-invertible
Received by editor(s): November 3, 2003
Posted: February 24, 2005
Communicated by: Bernd Ulrich
Copyright of article: Copyright 2005, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.




AMS and Social Media LinkedIn Facebook Podcasts Twitter YouTube RSS Feeds Blogs Wikipedia