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Ring extensions with some finiteness conditions on the set of intermediate rings

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Abstract

A ring extension R ⊆ S is said to be FO if it has only finitely many intermediate rings. R ⊆ S is said to be FC if each chain of distinct intermediate rings in this extension is finite. We establish several necessary and sufficient conditions for the ring extension R ⊆ S to be FO or FC together with several other finiteness conditions on the set of intermediate rings. As a corollary we show that each integrally closed ring extension with finite length chains of intermediate rings is necessarily a normal pair with only finitely many intermediate rings. We also obtain as a corollary several new and old characterizations of Prüfer and integral domains satisfying the corresponding finiteness conditions.

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Authors and Affiliations

  1. Department of Mathematics, College of Sciences, University of Sharjah, P.O.Box 27272, Sharjah, U.A.E.

    Ali Jaballah

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  1. Ali Jaballah
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Correspondence to Ali Jaballah.

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Jaballah, A. Ring extensions with some finiteness conditions on the set of intermediate rings. Czech Math J 60, 117–124 (2010). https://doi.org/10.1007/s10587-010-0002-x

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  • DOI: https://doi.org/10.1007/s10587-010-0002-x

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