Prüfer modules in filtration categories of semi-bricks

Abstract: Let $R$ be an associative unitary ring. A brick in the module category $R-Mod$ is a finitely generated module with a division ring as endomorphism ring. Two non-isomorphic bricks $X,Y$ are said to be orthogonal if $Hom(X,Y)=Hom(Y,X)=0$, and a class ${\mathcal X}$ of pairwise orthogonal bricks is called a semi-brick. We consider the full subcategory $Filt({\mathcal X})$ of modules with filtration in ${\mathcal X}$ and show that this category is a wide category. Let ${\mathcal X}^{\perp}$ be the class of modules $M$ with $Ext^1(X, M)=0$ for all $X\in {\mathcal X}$. Then for each module $Y\in R-Mod$ there exists a ${\mathcal X}^{\perp}$-envelope $Y_{{\mathcal X}}(\infty)$ which can be constructed as the direct limit of iterated universal short exact sequences. Let now in addition the projective dimension of every $X\in {\mathcal X}$ be lower or equal $1$. Then we call for $Y\in {\mathcal X}$ the ${\mathcal X}^{\perp}$-envelope a Pr\"ufer module since they share many properties with Pr\"ufer groups and also with Pr\"ufer modules over tame hereditary algebras. Their endomorphism rings are complete discrete valuation rings and every injective object in $Filt{{\mathcal X}}$ is isomorphic to a direct sum of Pr\"ufer modules. We apply the toolset to tame hereditary algebras and give an alternative proof for the classification of divisible modules. For wild hereditary algebras we show that they have large classes of semi-bricks.