Preorders on canonical families of modules of finite length

Abstract

Let R be an artinian ring. A family, Jt, of isomorphism types of /{-modules of finite length is said to be canonical if every /{-module of finite length is a direct sum of modules whose isomorphism types are in Jf . In this paper we show that Jt is canonical if the following conditions are simultaneously satisfied: (a) Jt contains the isomorphism type of every simple /{-module; (b) JK has a preorder with the property that every nonempty subfamily of J! 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 II, by a module of isomorphism type II 2 , with II,, II 2 in Jf, then M contains a submodule whose type II 3 is in Jt and II, does not precede II 3 . 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 /{-modules of finite length that satisfies Conditions (b) and (c).