Injective hulls with distinct ring structures

It is well known from Osofsky’s work that the injective hull E(R_R) of a ring R need not have a ring structure compatible with its R-module scalar multiplication. A closely related question is: if E(R_R) 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 R_R 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(R_R) also are ring structures on E(_RR).