J Austral Math Soc Ser A 53 pp64--77, 1992.

Preorders on Canonical Families of Modules of Finite Length

Uri Fixman and Frank Okoh

Abstract

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).

1991 AMS Subject Classification: 16D70, 15A21

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Authors

Uri Fixman

Queen's University, Kingston, Ontario K7L 3N6, Canada.

Frank Okoh

Wayne State University, Detroit, Michigan 48202, USA.