IDEAS home Printed from https://ideas.repec.org/a/spr/indpam/v51y2020i1d10.1007_s13226-020-0394-8.html
   My bibliography  Save this article

A Generalization of Posner’s Theorem on Derivations in Rings

Author

Listed:
  • Fuad Ali Ahmed Almahdi

    (King Khalid University)

  • Abdellah Mamouni

    (University Moulay Ismail)

  • Mohammed Tamekkante

    (University Moulay Ismail)

Abstract

In this paper, we generalize the Posner’s theorem on derivations in rings as follows: Let R be an arbitrary ring, P be a prime ideal of R, and d be a derivation of R. If [[x, d(x)], y] ∈ P for all x, y ∈ R, then d(R) ⊆ P or R/P is commutative. In particular, if R is semiprime and d is a centralizing derivation of R, we prove that either R is commutative or there exists a minimal prime ideal P of R such that d(R) ⊆ P. As a consequence, we show that for any semiprime ring with a centralizing derivation there exists at least a minimal prime ideal P such that d(P) ⊆ P.

Suggested Citation

  • Fuad Ali Ahmed Almahdi & Abdellah Mamouni & Mohammed Tamekkante, 2020. "A Generalization of Posner’s Theorem on Derivations in Rings," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(1), pages 187-194, March.
    • Handle: RePEc:spr:indpam:v:51:y:2020:i:1:d:10.1007_s13226-020-0394-8
    DOI: 10.1007/s13226-020-0394-8
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13226-020-0394-8
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13226-020-0394-8?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Moulay Abdallah Idrissi & Lahcen Oukhtite, 2022. "Structure of a quotient ring $$\pmb {R/P}$$ R / P with generalized derivations acting on the prime ideal P and some applications," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(3), pages 792-800, September.

    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:indpam:v:51:y:2020:i:1:d:10.1007_s13226-020-0394-8. 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.