Bulletin of the
Korean Mathematical Society BKMS

Let $R$ be a ring and $\CQ$ be a quiver. In this paper we give another definition of purity in the category of quiver representations. Under such definition we prove that the class of all pure injective representations of $\CQ$ by $R$-modules is preenveloping. In case $\CQ$ is a left rooted semi-co-barren quiver and $R$ is left Noetherian, we show that every cotorsion flat representation of $\CQ$ is pure injective. If, furthermore, $R$ is $n$-perfect and $\CF$ is a flat representation $\CQ$, then the pure injective dimension of $\CF$ is at most $n$.

Keywords: representation of quiver, pure monomorphism, pure injective representation, cotorsion representation, flat representation, pure injective resolution

MSC numbers: Primary 16G20; Secondary 16E10