We consider rings over which the injective hulls of simple modules are locally Artinian. We will give an outline of the problem and talk about recent developments. In particular, we will see when the injective hulls of simple modules over an Ore extension R = K[x][y;d] are locally Artinian. Specifically we will show that the injective hulls of simple modules over R are locally Artinian if and only if R is isomorphic to the first Weyl algebra or to the polynomial ring K[x,y].

* This is a joint work with Paula A.A.B. Carvalho and Christian Lomp.