Skip to main content
×
×
Home

Interlacing methods and large indecomposables

  • S. E. Dickson (a1) and G. M. Kelly (a2)
Abstract

The method of interlacing of modules, like amalgamation of groups, is a way of getting new objects from old. Briefly, the interlacing module we consider is a certain factor module of a direct sum of copies (finite or infinite) of an original module M. The conditions given in a previous paper by the first author in order that the interlacing module (using finitely many copies) be indecomposable are here greatly weakened, and we further allow the number of copies of the original to be infinite. R. Colby has shown that if R is a left artinian ring, the existence of a bound on the number of generators required for any indecomposable finitely-generated left R-module implies that R has a distributive lattice of two-sided ideals. This result is extended to rings whose identity is a sum of orthogonal local idempotents.

For these rings the same distributivity is proved in case every indecomposable interlacing module of the above type which begins with an indecomposable projective M is finitely-generated. A consequence is that any finite-dimensional algebra over a field having infinitely many two-sided ideals has infinite-dimensional indecomposables.

    • Send article to Kindle

      To send this article to your Kindle, first ensure no-reply@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 sending to your Kindle. Find out more about sending to your Kindle.

      Note you can select to send to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be sent 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.

      Interlacing methods and large indecomposables
      Available formats
      ×
      Send article to Dropbox

      To send this article to your Dropbox account, please select one or more formats and 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 <service> account. Find out more about sending content to Dropbox.

      Interlacing methods and large indecomposables
      Available formats
      ×
      Send article to Google Drive

      To send this article to your Google Drive account, please select one or more formats and 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 <service> account. Find out more about sending content to Google Drive.

      Interlacing methods and large indecomposables
      Available formats
      ×
Copyright
References
Hide All
[1]Brenner, Sheila, “Modular representations of p–groups”, J. Algebra 15 (1970), 89102.
[2]Butler, M.C.R., “On the structure of modules over certain augmented algebras”, Proc. London Math. Soc. (to appear).
[3]Colby, R.R., “On indecomposable modules over rings with minimum condition”, Pacific J. Math. 19 (1966), 2333.
[4]Corner, A.L.S., “Endomorphism algebras of large modules with distinguished submodules”, J. Algebra 11 (1969), 155185.
[5]Dickson, S.E., “On algebras of finite representation type”, Trans. Amer. Math. Soc. 135 (1969), 127141.
[6]Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachussets, 1966).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Metrics

Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed