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)



The content of a Gaussian polynomial is invertible

Authors: K. Alan Loper and Moshe Roitman
Journal: Proc. Amer. Math. Soc. 133 (2005), 1267-1271
MSC (2000): Primary 13B25
Published electronically: December 15, 2004
MathSciNet review: 2111931
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $R$ be an integral domain and let $f(X)$ be a nonzero polynomial in $R[X]$. The content of $f$ is the ideal $\mathfrak c(f)$ generated by the coefficients of $f$. The polynomial $f(X)$ is called Gaussian if $\mathfrak c(fg) = \mathfrak c(f)\mathfrak c(g)$ for all $g(X) \in R[X]$. It is well known that if $\mathfrak c(f)$ is an invertible ideal, then $f$ is Gaussian. In this note we prove the converse.

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

  • 1. 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)
  • 2. W. Bruns and A. Guerrieri, The Dedekind-Mertens formula and determinantal rings, Proc. Amer. Math. Soc. 127 (1999), 657-663. MR 1468185 (99f:13013)
  • 3. A. Corso, W. Heinzer, and C. Huneke, A generalized Dedekind-Mertens lemma and its converse, Trans. Amer. Math. Soc. 350 (1998), 5095-5109. MR 1473435 (99b:13012)
  • 4. A. Corso, W. Vasconcelos, and R. Villarreal, Generic Gaussian ideals, J. Pure Appl. Algebra 125 (1998), 117-127. MR 1600012 (98m:13014)
  • 5. P. Eakin and A. Sathaye, Prestable ideals, J. Algebra 41 (1976), 439-454. MR 0419428 (54:7449)
  • 6. R. Gilmer, Multiplicative Ideal Theory, Queen's Papers in Pure and Applied Mathematics, vol. 90 1992, Kingston, Ontario, Canada. MR 1204267 (93j:13001)
  • 7. S. Glaz and W. V. Vasconcelos, The Content of Gaussian Polynomials, J. Algebra 202 (1998), 1-9. MR 1614237 (99c:13003)
  • 8. W. Heinzer and C. Huneke, Gaussian polynomials and content ideals, Proc. Amer. Math. Soc. 125 (1997), 739-745. MR 1401742 (97e:13015)
  • 9. 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)

Similar Articles

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

Retrieve articles in all journals with MSC (2000): 13B25

Additional Information

K. Alan Loper
Affiliation: Department of Mathematics, Ohio State University-Newark, Newark, Ohio 43055

Moshe Roitman
Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel

Keywords: Content, Gaussian polynomial, invertible ideal, locally principal, prestable ideal
Received by editor(s): September 16, 2003
Published electronically: December 15, 2004
Additional Notes: The second author thanks the Mathematics Department of Ohio State University for its hospitality
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society