Gaussian polynomials and invertibility

Lucas, Thomas

doi:10.1090/S0002-9939-05-07977-3

Gaussian polynomials and invertibility
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Proc. Amer. Math. Soc. 133 (2005), 1881-1886 Request permission

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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