Injective Hulls of Simple Modules over Differential Operator Rings

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