Pathological Modules over Tame Rings

where q ^ 2 is a fixed integer. All such pathological modules exist for the free associative algebra k(x,y) in two (non-commuting) variables over a field k. They therefore occur also for any ^-algebra A for which there exists a full embedding of the module category k(X> yjS)\ into the module category A9)l. Indeed, in order to ensure the existence of pathological ^-modules we need only require that a full subcategory of X9ft is representation equivalent to k algebra which has only finitely many indecomposable modules, then every indecomposable module is finite-dimensional and every module is a direct sum of indecomposable ones. Thus, for such A, there are no pathological modules at all. In this paper we consider the remaining case of an algebra of tame representation type—that is, the case of an algebra A which has no full subcategory representation equivalent to ft y> 9JZ, but which has infinitely many indecomposable representations. An example of a tame algebra is k[t], the polynomial ring in one variable over the field k. For all known finite-dimensional tame algebras there is a full subcategory of y4-modules which is representation equivalent to kW9Jt. Now the question is: what kinds of pathological modules does a tame algebra A have? We shall show that, even for A = k[t], the answer is: all types of pathological modules. From this result for /c|7]-modules it follows that all known finite-dimensional tame algebras also have all types of pathological modules.