The content of a Gaussian polynomial is invertible

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 \mathfra...

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Bibliographic Details
Published in:Proceedings of the American Mathematical Society 2005-05, Vol.133 (5), p.1267-1271
Main Authors: Loper, K. Alan, Roitman, Moshe
Format: Article
Language:English
Subjects:
Algebra
College mathematics
Commutative rings and algebras
Exact sciences and technology
Mathematical integrals
Mathematical rings
Mathematical theorems
Mathematics
Polynomials
Sciences and techniques of general use
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Summary: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.
ISSN:0002-9939
1088-6826
DOI:10.1090/S0002-9939-04-07826-8