IDEAS home Printed from https://ideas.repec.org/h/spr/sprchp/978-1-4471-0039-3_8.html
   My bibliography  Save this book chapter

Rings without finiteness assumption

In: Further Algebra and Applications

Author

Listed:
  • P. M. Cohn

    (University College London, Department of Mathematics)

Abstract

For general rings there is naturally not as much structure theory as in the Artinian or Noetherian case. It is true that some of the same ethods can be used, e.g. the radical can be defined, semiprimitive rings can be xpressed as subdirect products of primitive rings etc., but these methods are less precise and they do not lead to a complete classification. For primitive rings a structure theorem can be proved using a general version of the density theorem; this is presented in Section 8.1 and applied in Section 8.2, while Section 8.3 deals with semiprimitive rings. So far we have taken the existence of a unit element for granted, but some work has been done on `rings without one' and we cast a brief glance at it in ection 8.4; we shall examine the case of simple rings and also see when the existence of a `one' follows from other assumptions. In Section 8.5 we study semiprime rings, and in Section 8.6 we present an analogue of Goldie's theorem for PI-rings. The final section, Section 8.7, takes a brief look at a natural generalization of principal ideal domains: free ideal rings.

Suggested Citation

  • P. M. Cohn, 2003. "Rings without finiteness assumption," Springer Books, in: Further Algebra and Applications, chapter 8, pages 309-342, Springer.
    • Handle: RePEc:spr:sprchp:978-1-4471-0039-3_8
    DOI: 10.1007/978-1-4471-0039-3_8
    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:978-1-4471-0039-3_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.