Injective hulls with distinct ring structures

It is well known from Osofsky’s work that the injective hull E(RR)E(RR) of a ring RR need not have a ring structure compatible with its RR-module scalar multiplication. A closely related question is: if E(RR)E(RR) has a ring structure and its multiplication extends its RR-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 RR with R=Q(R)R=Q(R) (where Q(R)Q(R) is the maximal right ring of quotients of RR) such that every injective hull of RRRR 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)E(RR) also are ring structures on E(RR)E(RR).