IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-3-031-28847-0_19.html
   My bibliography  Save this book chapter

Around Prüfer Extensions of Rings

In: Algebraic, Number Theoretic, and Topological Aspects of Ring Theory

Author

Listed:
  • Gabriel Picavet

    (Mathématiques)

  • Martine Picavet-L’Hermitte

    (Mathématiques)

Abstract

This chapter intends to apply the properties of Prüfer extensions, investigated in the Knebusch–Zhang book, to ring extensions R ⊆ S. The integral closure R ¯ $$\overline R$$ of R in S is shown to be the intersection of all T ∈ [R, S], such that T ⊆ S is Prüfer. We are then able to establish an avoidance lemma for integrally closed subextensions. Rings of sections of the affine scheme defined by R provide results on S-regular ideals. Some results on pullbacks characterizations of Prüfer extensions are given. We introduce locally strong divisors, examining the properties of strong divisors of a local ring and their links with Prüfer extensions. The locally strong divisors allow us to give characterizations of QR extensions. We then derive some results on minimal and FCP extensions. Finally, we study the set of all primitive elements in an extension.

Suggested Citation

  • Gabriel Picavet & Martine Picavet-L’Hermitte, 2023. "Around Prüfer Extensions of Rings," Springer Books, in: Jean-Luc Chabert & Marco Fontana & Sophie Frisch & Sarah Glaz & Keith Johnson (ed.), Algebraic, Number Theoretic, and Topological Aspects of Ring Theory, pages 351-382, Springer.
    • Handle: RePEc:spr:sprchp:978-3-031-28847-0_19
    DOI: 10.1007/978-3-031-28847-0_19
    as

    Download full text from publisher

    To our knowledge, this item is not available for download. To find whether it is available, there are three options:
    1. Check below whether another version of this item is available online.
    2. Check on the provider's web page whether it is in fact available.
    3. Perform a
    for a similarly titled item that would be available.

    More about this item

    Keywords

    ;
    ;
    ;
    ;
    ;
    ;

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:sprchp:978-3-031-28847-0_19. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    We have no bibliographic references for this item. You can help adding them by using this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.