Skip to main content Accessibility help
×
Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-06-13T08:54:15.889Z Has data issue: false hasContentIssue false

6 - Centralizers and subalgebras

Published online by Cambridge University Press:  22 August 2009

P. M. Cohn
Affiliation:
University College London
Get access

Summary

The first topic of this chapter is commutativity in firs. We shall find that any maximal commutative subring of a 2-fir with strong DFL is integrally closed (Corollary 1.2), and the same method allows us to describe the centres of 2-firs as integrally closed rings and make a study of invariant elements in 2-firs and their factors in Sections 6.1 and 6.2. The well-known result that a simple proper homomorphic image of a principal ideal domain is a matrix ring over a skew field is generalized here to atomic 2-firs (Theorem 2.4). In Section 6.3 the centres of principal ideal domains are characterized as Krull domains. Further, the centre of a non-principal fir is shown to be a field in Section 6.4.

Secondly we look at subalgebras and ideals of free algebras in Section 6.6; by way of preparation submonoids of free monoids are treated in section 6.5. A brief excursion into coding theory shows how the Kraft–McMillan inequality can be used to find free subalgebras, and the fir property of free algebras is again derived (Theorem 6.7). Section 6.7 is devoted to a fundamental theorem on free algebras: Bergman's centralizer theorem (Theorem 7.7).

Section 6.8 deals with invariants under automorphisms of free algebras, and Section 6.9 treats the Galois correspondence between automorphism groups and free subalgebras, as described by Kharchenko.

Type
Chapter
Information
Publisher: Cambridge University Press
Print publication year: 2006

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Save book to Kindle

To save this book to your Kindle, first ensure coreplatform@cambridge.org is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • Centralizers and subalgebras
  • P. M. Cohn, University College London
  • Book: Free Ideal Rings and Localization in General Rings
  • Online publication: 22 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542794.011
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • Centralizers and subalgebras
  • P. M. Cohn, University College London
  • Book: Free Ideal Rings and Localization in General Rings
  • Online publication: 22 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542794.011
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • Centralizers and subalgebras
  • P. M. Cohn, University College London
  • Book: Free Ideal Rings and Localization in General Rings
  • Online publication: 22 August 2009
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542794.011
Available formats
×