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Preorders on canonical families of modules of finite length

Part of: Basic linear algebra Modules, bimodules and ideals

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

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