The aim of this paper is to extend the structure theory for infinitely generated modules over tame hereditary algebras to the more general case of modules over concealed canonical algebras. Using tilting, we may assume that we deal with canonical algebras. The investigation is centered around the generic and the Pr\"{u}fer modules, and how other modules are determined by these modules.

MSC Classifications:

16D70, 16D90, 16G20, 16G60, 16G70 show english descriptions Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation
Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality
Representations of quivers and partially ordered sets
Representation type (finite, tame, wild, etc.)
Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers 16D70 - Structure and classification (except as in 16Gxx), direct sum decomposition, cancellation
16D90 - Module categories [See also 16Gxx, 16S90]; module theory in a category-theoretic context; Morita equivalence and duality
16G20 - Representations of quivers and partially ordered sets
16G60 - Representation type (finite, tame, wild, etc.)
16G70 - Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers

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