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Preservation of the properties of the polynomial ring of the power series ring in the class of finite conductor rings. (Spanish. English summary) Zbl 1093.13019

It is known that properties of a ring \(R\) are sometimes inherited by both the polynomial ring \(R[x]\) and the power series ring \(R[[x]]\), sometimes by only one of them and sometimes by neither. This is the motivation for the study in this paper. The paper focuses on the preservation of properties such as \(GCD\), \(G-GCD\), coherence and quasi-coherence within the class of finite conductor rings (i.e. rings such that the intersection of two principal ideals is a finitely generated ideal).
Let \(R\) be an integral domain with quotient field \(K\). It is shown that \(R\) is \(GCD\) if and only if \(R[x]\) is a \(GCD\) domain. \(R\) is said to be a \(G-GCD\) domain if the intersection of two invertible integral ideals is again an invertible integral ideal. \(R\) being a \(G-GCD\) domain is shown to be equivalent to \(R[x]\) being a \(G-GCD\) domain or to \(R(x)=R[x]S^{-1}\) being a \(GCD\) domain, where \(S=\{f \in R[x] | c(f)=R\}\) and \(c(f)\) denotes the fractional ideal generated by the coefficients of \(f\). A coherent (resp. quasi-coherent) integrally closed domain is shown to have a quasi coherent (resp. finite conductor) polynomial ring \(R[x]\).
With respect to power series rings, for an integral domain \(D\), \(D[[x]]\) is completely integrally closed if and only if \(D\) is completely integrally closed. Results showing that the property of \(GCD\) is not preserved by power series rings are also proven.

MSC:

13F20 Polynomial rings and ideals; rings of integer-valued polynomials
13F25 Formal power series rings
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