Residually FCP extensions of commutative rings

Article

Abstract

A ring extension \(R\subseteq S\) is said to be residually FCP if, for all \(Q\in \mathrm {Spec}(S)\), each chain of rings between \(R/(Q\cap R)\) and S / Q is finite. The aim of this note is to establish several necessary and sufficient conditions for such extensions. We also study them, with emphasis on base rings R of Krull dimension 0. In particular we show that if \(R\subseteq S\) is residually FCP and \(\dim (R)=0\), then S is integral over R. This generalizes a result due to Anderson and Dobbs (Math. Rep. (Bucur.) 3(53):95–103, 2001).

Keywords

Intermediate ring Minimal ring extension Maximal chain FCP and FIP Finite fibers INC 

Mathematics Subject Classification

13B02 13A15 13B21 13E05 

Notes

Acknowledgements

The author thanks the referee for his several helpful remarks concerning the final form of this paper.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Faculty of Sciences, Department of MathematicsSfax UniversitySfaxTunisia

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