AbstractIt is well known from Osofsky’s work that the injective hull E(RR) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(RR) has a ring structure and its multiplication extends its R-module scalar multiplication, must the ring structure be unique? In this paper, we utilize the properties of Morita duality to explicitly describe an injective hull of a ring R with R=Q(R) (where Q(R) is the maximal right ring of quotients of R) such that every injective hull of RR has (possibly infinitely many) distinct compatible ring structures which are mutually ring isomorphic and quasi-Frobenius. Further, these rings have the property that the ring structures for E(RR) al...