Pruefer modules in filtration categories of semibricks

Let $R$ be a ring with unity and $\mathcal{X}$ a semibrick in the module category $\mathrm{M}\mathrm{o}\mathrm{d}R$, that is, a class of pairwise orthogonal finitely presented modules whose endomorphism rings are division rings. We study the full subcategory $\mathrm{F}\mathrm{i}\mathrm{l}\mathrm{t}(\mathcal{X})$ consisting of all modules admitting a filtration with factors in $\mathcal{X}$. We show that $\mathrm{F}\mathrm{i}\mathrm{l}\mathrm{t}(\mathcal{X})$ is a wide subcategory of $\mathrm{M}\mathrm{o}\mathrm{d}R$. For the Ext-orthogonal class $${\mathcal{X}}^{\perp }=\{M\in \mathrm{M}\mathrm{o}\mathrm{d}R\mid {\mathrm{Ext}}_R^1(X,M)=0\text{ for all }X\in \mathcal{X}\}$$ we construct, for every module $Y$, an ${\mathcal{X}}^{\perp }$-envelope $Y_{\mathcal{X}}(\mathrm{\infty })$ as a direct limit of iterated universal short exact sequences. Assume that every $X\in \mathcal{X}$ has projective dimension at most one and that ${\mathrm{Hom}}_R(X,R)=0$ for all $X\in \mathcal{X}$. Then the envelope $R_{\mathcal{X}}(\mathrm{\infty })$ of the regular module is isomorphic to the universal localization $R_{\mathcal{X}}$ of $R$ at $\mathcal{X}$ in the sense of Schofield. The ${\mathcal{X}}^{\perp }$-envelopes of modules in $\mathcal{X}$ are called Prüfer modules since they share many properties with classical Prüfer groups and with Prüfer modules over tame hereditary algebras. We prove that every injective object in $\mathrm{F}\mathrm{i}\mathrm{l}\mathrm{t}(\mathcal{X})$ is a direct sum of such Prüfer modules.