Gaussian Property of the Rings R(X) and R〈X

The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (...

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Bibliographic Details
Published in:Communications in algebra 2016-04, Vol.44 (4), p.1636-1646
Main Authors: McGovern, Warren WM, Sharma, Madhav
Format: Article
Language:English
Subjects:
Arithmetical ring
Gaussian ring
Nagata ring
Prüfer ring
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Summary:The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. A commutative ring R is said to be Gaussian if c(fg) = c(f)c(g) for every polynomials f and g in R[X]. A number of authors have formulated necessary and sufficient conditions for R(X) (respectively, R⟨X⟩) to be semihereditary, have weak global dimension at most one, be arithmetical, or be Prüfer. An open question raised by Glaz is to formulate necessary and sufficient conditions that R(X) (respectively, R⟨X⟩) have the Gaussian property. We give a necessary and sufficient condition for the rings R(X) and R⟨X⟩ in terms of the ring R in case the square of the nilradical of R is zero.
ISSN:0092-7872
1532-4125
DOI:10.1080/00927872.2015.1027371