Residually FCP extensions of commutative rings



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).


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

Mathematics Subject Classification

13B02 13A15 13B21 13E05 



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


  1. 1.
    Anderson, D.D., Dobbs, D.E., Mullins, B.: The primitive element theorem for commutative algebras. Houst. J. Math. 25, 603–623 (1999)MathSciNetMATHGoogle Scholar
  2. 2.
    Anderson, D.D., Dobbs, D.E.: Residually FIP extensions of commutative rings. Math. Rep. (Bucur.) 3(53), 95–103 (2001)MathSciNetMATHGoogle Scholar
  3. 3.
    Ayache, A., Dobbs, D.E.: Finite maximal chains of commutative rings. J. Algebr. Appl. 14(1), 1450075 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Ayache, A., Jaballah, A.: Residually algebraic pairs of rings. Math. Z. 225, 49–65 (1997)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bourbaki, N.: Commutative Algebra. Addison-Wesley, Reading (1972)MATHGoogle Scholar
  6. 6.
    Dobbs, D.E.: On INC-extensions and polynomials with unit content. Can. J. Math. 33, 37–42 (1980)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Dobbs, D.E., Mullins, B., Picavet, G., Picavet-L’Hermitte, M.: On the FIP property for extensions of commutative rings. Comm. Algebr. 33, 3091–3119 (2005)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dobbs, D.E., Mullins, B., Picavet, G., Picavet-L’Hermitte, M.: On the maximal cardinality of chains of intermediate rings. Int. Electron. J. Algebr. 5, 121–134 (2009)MathSciNetMATHGoogle Scholar
  9. 9.
    Dobbs, D.E., Picavet, G., Picavet-L’Hermitte, M.: Characterizing the ring extensions that satisfy FIP or FCP. J. Algebr. 371, 391–429 (2012)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Dobbs, D.E., Picavet, G., Picavet-L’Hermitte, M.: Transfer results for the FIP and FCP properties of ring extensions. J. Algebr. 43, 1279–1316 (2015)MathSciNetMATHGoogle Scholar
  11. 11.
    Gilmer, G.: Some finiteness conditions on the set of overrings of an integral domain. Proc. Am. Math. Soc. 131, 2337–2346 (2002)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Picavet, G., Picavet-L’Hermitte, M.: FIP and FCP products of ring morphisms. Palest. J. Math. 5(Spec1), 63–80 (2016)MathSciNetMATHGoogle Scholar
  13. 13.
    Jaballah, A.: Finiteness of the set of intermediary rings in normal pairs. Saitama Math. J. 17, 59–61 (1999)MathSciNetMATHGoogle Scholar
  14. 14.
    Jaballah, A.: Ring extensions with some finiteness conditions on the set of intermediate rings. Czechoslov. Math. J. 60(135), 117–124 (2010)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kaplansky, I.: Commutative Rings, Rev. edn. University of Chicago Press, Chicago (1974)MATHGoogle Scholar
  16. 16.
    Papick, I.J.: Topologically defined classes of going-down rings. Trans. Am. Math. Soc. 219, 1–37 (1976)CrossRefMATHGoogle Scholar

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© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

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

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