Elsevier

Journal of Algebra

Volume 99, Issue 1, March 1986, Pages 263-274
Journal of Algebra

Kronecker function rings and abstract Riemann surfaces

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Abstract

Let R be an integral domain, X(R) the abstract Riemann surface of R, and (R′)b the Kronecker function ring of the integral closure R′ of R. It is proved that there exists a homeomorphism, natural in R, between X(R) and Spec((R′)b). Ideal-theoretic and topological results are given for the extension j:R(R)b, notably that R is a Prüfer domain if and only if R = R′ and j is universally going-down. It is also proved that each spectral space X is a closed spectral image of a treed spectral space Y; if X is irreducible, Y can be taken as an abstract Riemann surface.

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Supported in part by the University of Tennessee and the Universitá di Roma “La Sapienza.”

Work done under the auspices of the GNSAGA of the Consiglio Nazionale delle Ricerche.