Submodule categories of wild representation type
Let Lambda be a commutative local uniserial ring with radical factor field k. We consider the category S(Lambda) of embeddings of all possible submodules of finitely generated Lambda-modules. In case Lambda=Z/(p(n)), where p is a prime, the problem of classifying the objects in S(Lambda), up to isomorphism, has been posed by Garrett Birkhoff in 1934. In this paper we assume that A has Loewy length at least seven. We show that S(A) is controlled k-wild with a single control object I is an element of S(Lambda). It follows that each finite dimensional k-algebra can be realized as a quotient End(X)/End(X)(I) of the endomorphism ring of some object X is an element of S(Lambda) modulo the ideal End(X)(I) of all maps which factor through a finite direct sum of copies of I. (c) 2005 Elsevier B.V. All rights reserved.
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Elsevier Science