Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-23T02:24:42.567Z Has data issue: false hasContentIssue false
  • English
  • Français

Interlacing methods and large indecomposables

Published online by Cambridge University Press:  17 April 2009

S. E. Dickson  and
G. M. Kelly
S. E. Dickson
Affiliation:
University of Newcastle, Newcastle, New South Wales, and Iowa State University, Ames, Iowa, USA;
G. M. Kelly
Affiliation:
University of New South Wales, Kensington, New South Wales.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 3 , Issue 3 , December 1970 , pp. 337 - 348
Copyright
Copyright © Australian Mathematical Society 1970

References

[1]Brenner, Sheila, “Modular representations of p–groups”, J. Algebra 15 (1970), 89–102.CrossRefGoogle Scholar
[2]Butler, M.C.R., “On the structure of modules over certain augmented algebras”, Proc. London Math. Soc. (to appear).Google Scholar
[3]Colby, R.R., “On indecomposable modules over rings with minimum condition”, Pacific J. Math. 19 (1966), 23–33.CrossRefGoogle Scholar
[4]Corner, A.L.S., “Endomorphism algebras of large modules with distinguished submodules”, J. Algebra 11 (1969), 155–185.CrossRefGoogle Scholar
[5]Dickson, S.E., “On algebras of finite representation type”, Trans. Amer. Math. Soc. 135 (1969), 127–141.CrossRefGoogle Scholar
[6]Lambek, J., Lectures on rings and modules (Blaisdell, Waltham, Massachussets, 1966).Google Scholar