Injective hulls of simple modules over differential operator rings

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.