Abstract
The focus of this chapter is on right essential overrings of a ring which are not right rings of quotients. Osofsky’s well-known example of a finite ring whose injective hull has no compatible ring structure is considered and generalized. All possible right essential overrings of the ring in Osofsky’s example are discussed. A ring R is constructed with a module essential extension S which is not the injective hull of R. However, S is shown to have one compatible ring structure which is a QF-ring and another compatible ring structure which is not even right FI-extending. Finally, Osofsky compatibility is discussed and a class of rings whose injective hulls have distinct compatible ring structures is studied.
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