Relative homological algebra in categories of representations of infinite quivers

Abstract

In the first part of this paper, we prove the existence of torsion free covers in the category of representations of quivers, $(Q,R-Mod)$, for a wide class of quivers included in the class of the so-called source injective representation quivers provided that any direct sum of torsion free and injective $R$-modules is injective. In the second part, we prove the existence of $\mathscr{F}_{cw}$-covers and $\mathscr{F}_{cw}^{\perp}$-envelopes for any quiver $Q$ and any ring $R$ with unity, where $\mathscr{F}_{cw}$ is the class of all "componentwise" flat representations of $Q$.