  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.
