Let $A$ be a unitary artinian ring. Suppose additionally that the injective hulls of all simple modules have finite length (or, equivalently, are finitely generated).
Does the ring $A$ have to possess a Morita self-duality? Is there any such ring $A$ which is not an Artin algebra?
In order to try to answer to this I looked at "Finiteness of the injective hull", Rosenberg and Zelinski; "The dual notion of "finitely generated"", Vámos; "On co-noetherian rings", Jens. The paper "Co-Artinian rings and Morita duality", Miller and Turnidge, may be useful but I don't have access to it.