Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-24T17:24:45.755Z Has data issue: false hasContentIssue false
  • English
  • Français

Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras

Published online by Cambridge University Press:  20 November 2018

F. Okoh
F. Okoh*
Affiliation:
Department of Mathematics, Wayne State University, Detroit, Michigan 48202, U.S.A.
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.

If R is a Dedekind domain, then div splits i.e.; the maximal divisible submodule of every R-module M is a direct summand of M. We investigate the status of this result for some finite-dimensional hereditary algebras. We use a torsion theory which permits the existence of torsion-free divisible modules for such algebras. Using this torsion theory we prove that the algebras obtained from extended Coxeter- Dynkin diagrams are the only such hereditary algebras for which div splits. The field of rational functions plays an essential role. The paper concludes with a new type of infinite-dimensional indecomposable module over a finite-dimensional wild hereditary algebra.

Keywords

16D7016G6013C12
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 39 , Issue 1 , 01 March 1996 , pp. 111 - 114
Copyright
Copyright © Canadian Mathematical Society 1995

References

1. Aronszajn, N. and Fixman, U., Algebraic spectral problems, Studia Mathematica 30(1968), 273–338.Google Scholar
2. Britten, D. and Lemire, F., On pointed modules of simple Lie algebras, CMS conference proceedings 5, 319-323.Google Scholar
3. Coleman, A. J. and Futorny, V., Stratified L-modules, J. Algebra, 163(1994), 219–234.Google Scholar
4. Dlab, V. and Ringel, C. M., Indecomposable representations of graphs and algebras, Memoirs Amer. Math. Soc. 173(1976).Google Scholar
5. Fixman, U., Okoh, F., and Sankaran, N., Internal functors for systems of linear transformations, J. Algebra, (1988)399-415.Google Scholar
6. Jensen, C. U. and Lenzing, H., Model Theoretic Algebra, Gordon and Breach, New York and London, 1989.Google Scholar
7. Kaplansky, I., Modules over Dedekind domains and valuation rings, Trans. Amer. Math. Soc. 72(1952), 327–340.Google Scholar
8. Kerner, O., Preprojective components of wild tilted algebras, Manuscripta Math. 61 ( 1988), 429–445.Google Scholar
9. Levy, L., Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15(1963), 132— 151.Google Scholar
10. Lukas, F., Infinite-dimensional modules over wild hereditary algebras, J. London Math. Soc.(2)14(1991), 401–419.Google Scholar
11. Okoh, F., Properties of purely simple Kronecker modules, Journ. Pure and Applied Alg. (1983), 39-48.Google Scholar
12. Okoh, F., Pure-injective modules over path algebras, Journ. Pure and Applied Alg. (1991), 75-83.Google Scholar
13. Ringel, C. M., Infinite-dimensional representations of finite-dimensional hereditary algebras, Sympos. Math. Inst. Alta. Mat. 23(1979), 321–412.Google Scholar