  Let $L_K(E)$ denote the Leavitt path algebra associated to the finite graph
$E$ and field $K$. For any closed path $c$ in $E$, we define and investigate
the uniserial, artinian, non-noetherian left $L_K(E)$-module $U_{E,c-1}$. The
unique simple factor of each proper submodule of $U_{E,c-1}$ is isomorphic to
the Chen simple module $V_{[c^\infty]}$. In our main result, we classify those
closed paths $c$ for which $U_{E,c-1}$ is injective. In this situation,
$U_{E,c-1}$ is the injective hull of $V_{[c^\infty]}$.
