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Preorders on canonical families of modules of finite length

Part of: Modules, bimodules and ideals Basic linear algebra

Published online by Cambridge University Press:  09 April 2009

Uri Fixman  and
Frank Okoh
Uri Fixman
Affiliation:
Queen's UniversityKingston, Ontario K7L 3N6, Canada
Frank Okoh
Affiliation:
Wayne State UniversityDetroit, Michigan 48202, USA
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Abstract

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Let R be an artinian ring. A family, ℳ, of isomorphism types of R-modules of finite length is said to be canonical if every R-module of finite length is a direct sum of modules whose isomorphism types are in ℳ. In this paper we show that ℳ is canonical if the following conditions are simultaneously satisfied: (a) ℳ contains the isomorphism type of every simple R-module; (b) ℳ has a preorder with the property that every nonempty subfamily of ℳ with a common bound on the lengths of its members has a smallest type; (c) if M is a nonsplit extension of a module of isomorphism type II1 by a module of isomorphism type II2, with II1, II2 in ℳ, then M contains a submodule whose type II3 is in ℳ and II1 does not precede II3. We use this result to give another proof of Kronecker's theorem on canonical pairs of matrices under equivalence. If R is a tame hereditary finite-dimensional algebra we show that there is a preorder on the family of isomorphism types of indecomposable R-modules of finite length that satisfies Conditions (b) and (c).

MSC classification

Secondary: 16D70: Structure and classification (except as in ), direct sum decomposition, cancellation 15A21: Canonical forms, reductions, classification
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 53 , Issue 1 , August 1992 , pp. 64 - 77
Copyright
Copyright © Australian Mathematical Society 1992

References

[1]Aronszajn, N., ‘Quadratic forms on vector spaces’, in: Proc. Intern. Symposium on Linear spaces, 1960, (Jerusalem Academic Press, Israel Academy of Sciences and Humanities, and Pergamon Press, London. Jerusalem 1961), pp. 29–87.Google Scholar
[2]Aronszajn, N. and Brown, R. D., ‘Finite-dimensional perturbations of spectral problems and variational approximation methods for eigenvalue problems I,’ Studia Math. 36 (1970), 1–76.CrossRefGoogle Scholar
[3]Aronszajn, N. and Fixman, U., ‘Algebraic spectral problems,’ Studia Math. 30 (1968), 273–338.CrossRefGoogle Scholar
[4]Curtis, C. W. and Reiner, I., Methods of representation theory with applications to finite groups and orders, Volume I, (John Wiley and Sons, New York, 1981).Google Scholar
[5]Dickson, L. E., ‘Singular case of bilinear, quadratic or hermitian forms,’ Trans. Amer. Math. Soc. 29 (1927), 239–253.CrossRefGoogle Scholar
[6]Dieudonné, J., ‘Sur la réduction canonique des couples de matrices,’ Bull. Soc. Math. France 74 (1946), 130–146.CrossRefGoogle Scholar
[7]Fixman, U., Okoh, F., and Sankaran, N., ‘Internal functors for systems of linear transformations,’ J. Algebra 113 (1988), 399–415.CrossRefGoogle Scholar
[8]Friedland, S., ‘Simultaneous similarity of matrices,’ Advances in Math. 50 (1983), 189–265.CrossRefGoogle Scholar
[9]Fuchs, L., Infinite abelian groups, Volume I, Academic Press, New York, 1970.Google Scholar
[10]Gantmacher, F. R., Theory of matrices (Interscience, New York, 1959.)Google Scholar
[11]Greither, C., ‘Die unzerlegbaren Moduln über K[X, Y]/(X, Y)2,’ Bericht Nr. 39, 1979, Seminar F. Kasch, B. Pareigis, Mathematisches Institut Der Universität München, Verlag Uni-Drück München.Google Scholar
[12]Heller, A. and Reiner, I., ‘Indecomposable representations,’ Illinois J. Math. 5 (1961), 314–323.CrossRefGoogle Scholar
[13]Jacobson, N., Lectures in abstract algebra, Volume, II Linear Algebra (D. Van Nostrand), Princeton, 1953.CrossRefGoogle Scholar
[14]Kronecker, L., ‘Algebraische Reduktion der Schaaren bilinearer Formen,’ Sitzungsberichte Akad. Berlin (1890), 1225–1237.Google Scholar
[15]Ringel, C. M., ‘Infinite-dimensional representations of finite-dimensional hereditary algebras,’ Symposia Mat. Inst. Alta Mat. 23 (1979), 321–412.Google Scholar
[16]Ringel, C. M., ‘Representation of K-species and bimodules,’ J. Algebra 41 (1976), 269–302.CrossRefGoogle Scholar
[17]Turnbull, H. W., ‘On the reduction of singular matrix pencils,’ Proc. Edinburgh Math. Soc. 4 (1935), 67–76.CrossRefGoogle Scholar
[18]Turnbull, H. W. and Aitken, A. C., An introduction to the theory of canonical matrices (Blackie, London, 1932).Google Scholar