Geodesic
Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras
Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 111-114

Voir la notice de l'article provenant de la source Cambridge University Press

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.
DOI : 10.4153/CMB-1996-014-9
Mots-clés : 16D70, 16G60, 13C12 
Okoh, F. Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras. Canadian mathematical bulletin, Tome 39 (1996) no. 1, pp. 111-114. doi: 10.4153/CMB-1996-014-9
@article{10_4153_CMB_1996_014_9,
     author = {Okoh, F.},
     title = {Torsion-Free and {Divisible} {Modules} {Over} {Finite-Dimensional} {Algebras}},
     journal = {Canadian mathematical bulletin},
     pages = {111--114},
     year = {1996},
     volume = {39},
     number = {1},
     doi = {10.4153/CMB-1996-014-9},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-014-9/}
}
TY  - JOUR
AU  - Okoh, F.
TI  - Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras
JO  - Canadian mathematical bulletin
PY  - 1996
SP  - 111
EP  - 114
VL  - 39
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-014-9/
DO  - 10.4153/CMB-1996-014-9
ID  - 10_4153_CMB_1996_014_9
ER  - 
%0 Journal Article
%A Okoh, F.
%T Torsion-Free and Divisible Modules Over Finite-Dimensional Algebras
%J Canadian mathematical bulletin
%D 1996
%P 111-114
%V 39
%N 1
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1996-014-9/
%R 10.4153/CMB-1996-014-9
%F 10_4153_CMB_1996_014_9

[1] 1. Aronszajn, N. and Fixman, U., Algebraic spectral problems, Studia Mathematica 30(1968), 273–338. Google Scholar

[2] 2. Britten, D. and Lemire, F., On pointed modules of simple Lie algebras, CMS conference proceedings 5, 319-323. Google Scholar

[3] 3. Coleman, A. J. and Futorny, V., Stratified L-modules, J. Algebra, 163(1994), 219–234. Google Scholar

[4] 4. Dlab, V. and Ringel, C. M., Indecomposable representations of graphs and algebras, Memoirs Amer. Math. Soc. 173(1976). Google Scholar

[5] 5. Fixman, U., Okoh, F., and Sankaran, N., Internal functors for systems of linear transformations, J. Algebra, (1988)399-415. Google Scholar

[6] 6. Jensen, C. U. and Lenzing, H., Model Theoretic Algebra, Gordon and Breach, New York and London, 1989. Google Scholar

[7] 7. Kaplansky, I., Modules over Dedekind domains and valuation rings, Trans. Amer. Math. Soc. 72(1952), 327–340. Google Scholar

[8] 8. Kerner, O., Preprojective components of wild tilted algebras, Manuscripta Math. 61 ( 1988), 429–445. Google Scholar

[9] 9. Levy, L., Torsion-free and divisible modules over non-integral domains, Canad. J. Math. 15(1963), 132— 151. Google Scholar

[10] 10. Lukas, F., Infinite-dimensional modules over wild hereditary algebras, J. London Math. Soc.(2)14(1991), 401–419. Google Scholar

[11] 11. Okoh, F., Properties of purely simple Kronecker modules, Journ. Pure and Applied Alg. (1983), 39-48. Google Scholar

[12] 12. Okoh, F., Pure-injective modules over path algebras, Journ. Pure and Applied Alg. (1991), 75-83. Google Scholar

[13] 13. Ringel, C. M., Infinite-dimensional representations of finite-dimensional hereditary algebras, Sympos. Math. Inst. Alta. Mat. 23(1979), 321–412. Google Scholar

Cité par Sources :