Pr\"ufer modules over wild hereditary algebras

Let A be a hereditary finite dimensional algebra over an algebraically closed field k. A brick is defined as a finitely generated module with k as endomorphism ring. Two non-isomorphic bricks $X,Y$ are said to be orthogonal if $Hom(X,Y)=Hom(Y,X)=0$. In this paper we show that a class X of pairwise orthogonal bricks allows to construct Pr\"ufer modules. We consider the category Filt(X) of modules with a filtration in X and show that Filt(X) has enough injective objects. We can construct them by an iteration of the Bongarz construction for an universal short exact sequence. We call the infinite dimensional, indecomposable injective objects in Filt(X) Pr\"ufer modules and show that they share many properties with the Pr\"ufer modules over tame hereditary algebras as defined by C. M. Ringel [11]. The results of this paper can also be applied to tame hereditary algebras. The construction gives a strict filtration of the generic module Q. We give an alternative proof for the classification of torsion-free divisible modules to show how useful this filtration is.