Injective hulls with distinct ring structures

Abstract

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 ( R R ) .