Composition factors for indecomposable modules

Author:
Dieter Happel

Journal:
Proc. Amer. Math. Soc. **86** (1982), 29-31

MSC:
Primary 16A64; Secondary 16A46

DOI:
https://doi.org/10.1090/S0002-9939-1982-0663860-4

MathSciNet review:
663860

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Abstract: Let be a field and be a finite-dimensional algebra over having only a finite number of isomorphism classes of indecomposable -modules. Let , be two indecomposable -modules. Then a homomorphism is said to be irreducible if for every factorization , is split mono or is split epi [**2**]. The aim of this note is to give an elementary proof of the fact that the indecomposable -modules are completely determined, up to isomorphism, by their composition factors if there is no chain of irreducible maps from an indecomposable module to itself. This theorem was first proved in [**5**] involving the theory of tilted algebras.

**[1]**Maurice Auslander,*Functors and morphisms determined by objects*, Representation theory of algebras (Proc. Conf., Temple Univ., Philadelphia, Pa., 1976) Dekker, New York, 1978, pp. 1–244. Lecture Notes in Pure Appl. Math., Vol. 37. MR**0480688****[2]**Maurice Auslander and Idun Reiten,*Representation theory of Artin algebras. IV. Invariants given by almost split sequences*, Comm. Algebra**5**(1977), no. 5, 443–518. MR**0439881**, https://doi.org/10.1080/00927877708822180**[3]**R. Bautista and F. Larion,*Auslander-Reiten quivers for certain algebras of finite representation type*(to appear).**[4]**Klaus Bongartz and Claus Michael Ringel,*Representation-finite tree algebras*, Representations of algebras (Puebla, 1980) Lecture Notes in Math., vol. 903, Springer, Berlin-New York, 1981, pp. 39–54. MR**654702****[5]**Dieter Happel and Claus Michael Ringel,*Tilted algebras*, Trans. Amer. Math. Soc.**274**(1982), no. 2, 399–443. MR**675063**, https://doi.org/10.1090/S0002-9947-1982-0675063-2

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DOI:
https://doi.org/10.1090/S0002-9939-1982-0663860-4

Article copyright:
© Copyright 1982
American Mathematical Society