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Rings, Polynomials, and Modules pp 129–143Cite as

Commutative Rings Whose Finitely Generated Ideals are Quasi-Flat

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Abstract

A definition of quasi-flat left module is proposed and it is shown that any left module which is either quasi-projective or flat is quasi-flat. A characterization of local commutative rings for which each ideal is quasi-flat (resp. quasi-projective) is given. It is also proven that each commutative ring R whose finitely generated ideals are quasi-flat is of λ-dimension ≤ 3, and this dimension ≤ 2 if R is local. This extends a former result about the class of arithmetical rings. Moreover, if R has a unique minimal prime ideal, then its finitely generated ideals are quasi-projective if they are quasi-flat.

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Notes

  1. 1.

    The module property M-flat is generally used to define flat module.

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Acknowledgements

At the Conference on Rings and Polynomials held in Graz (Austria), 3–8 July 2016, I did a talk entitled “Finitely generated powers of prime ideals”. An article with this title is appeared in Palest. J. Math., vol. 6(2). I thank again the organizers of this conference.

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Authors and Affiliations

  1. Normandie Université, F14032, Caen, France

    François Couchot

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  1. François Couchot
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Correspondence to François Couchot .

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Editors and Affiliations

  1. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Marco Fontana

  2. Department of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Sophie Frisch

  3. Department of Mathematics, University of Connecticut, Storrs, Connecticut, USA

    Sarah Glaz

  4. Dipartimento di Matematica e Fisica, Università degli Studi Roma Tre, Rome, Italy

    Francesca Tartarone

  5. Dipartimento di Matematica, Università di Padova, Padova, Italy

    Paolo Zanardo

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Couchot, F. (2017). Commutative Rings Whose Finitely Generated Ideals are Quasi-Flat. In: Fontana, M., Frisch, S., Glaz, S., Tartarone, F., Zanardo, P. (eds) Rings, Polynomials, and Modules. Springer, Cham. https://doi.org/10.1007/978-3-319-65874-2_7

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