Two characterizations of pure injective modules

Abstract

Let $R$ be a commutative ring with identity and $D$ an $R$-module. It is shown that if $D$ is pure injective, then $D$ is isomorphic to a direct summand of the direct product of a family of finitely embedded modules. As a result, it follows that if $R$ is Noetherian, then $D$ is pure injective if and only if $D$ is isomorphic to a direct summand of the direct product of a family of Artinian modules. Moreover, it is proved that $D$ is pure injective if and only if there is a family $\{T_\lambda\}_{\lambda\in \Lambda}$ of $R$-algebras which are finitely presented as $R$-modules, such that $D$ is isomorphic to a direct summand of a module of the form $\Pi_{\lambda\in \Lambda}E_\lambda$ where for each $\lambda\in \Lambda$, $E_\lambda$ is an injective $T_\lambda$-module.