  We study injective hulls of simple modules over differential operator rings
$R[\theta; d]$, providing necessary conditions under which these modules are
locally Artinian. As a consequence we characterize Ore extensions of
$S=K[x][\theta;\sigma, d]$ for $d$ a $K$-linear derivation and $\sigma$ a
$K$-linear automorphism of $K[x]$ such that injective hulls of simple
$S$-modules are locally Artinian.
