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Proceedings of the American Mathematical Society
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
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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.


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Additional Information

K. Alan Loper
Affiliation: Department of Mathematics, Ohio State University-Newark, Newark, Ohio 43055
Email: lopera@math.ohio-state.edu

Moshe Roitman
Affiliation: Department of Mathematics, University of Haifa, Haifa 31905, Israel
Email: mroitman@math.haifa.ac.il

DOI: http://dx.doi.org/10.1090/S0002-9939-04-07826-8
PII: S 0002-9939(04)07826-8
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.