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Gaussian polynomials and invertibility


Author: Thomas G. Lucas
Journal: Proc. Amer. Math. Soc. 133 (2005), 1881-1886
MSC (2000): Primary 13A15, 13B25
DOI: https://doi.org/10.1090/S0002-9939-05-07977-3
Published electronically: February 24, 2005
MathSciNet review: 2137851
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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 $c(f)Hom_R(c(f),R)=R$.


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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: https://doi.org/10.1090/S0002-9939-05-07977-3
Keywords: Gaussian polynomial, content, $Q_0$-invertible
Received by editor(s): November 3, 2003
Published electronically: February 24, 2005
Communicated by: Bernd Ulrich
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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