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Residually FCP extensions of commutative rings

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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).

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Acknowledgements

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

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  1. Faculty of Sciences, Department of Mathematics, Sfax University, B.P. 1171, 3000, Sfax, Tunisia

    Nabil Zeidi

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  1. Nabil Zeidi
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Correspondence to Nabil Zeidi.

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Communicated by M.Fontana.

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Zeidi, N. Residually FCP extensions of commutative rings. Ricerche mat 68, 375–381 (2019). https://doi.org/10.1007/s11587-018-0409-5

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  • DOI: https://doi.org/10.1007/s11587-018-0409-5

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