Approximately forty-five years ago K. Morita presented the first major results on equivalences and dualities between categories of modules over a pair of rings. These results, which characterized an equivalence between the entire categories of right (or left) modules over two rings as being represented by the covariant Hom and tensor functors induced by a balanced bimodule that is a so-called progenerator on either side, and which characterized a duality between reasonably large subcategories of right and left modules over two rings as being represented by the contravariant Hom functors induced by a balanced bimodule that is an injective cogenerator on both sides, have come to be known as the Morita theorems.

Morita's theorems on equivalence are exemplified by the equivalence of the categories of right modules over a simple artinian ring and the right vector spaces over its underlying division ring. More than a dozen years later the second author, expanding on the relationship between the category of modules generated by a simple module and the vector spaces over its endomorphism ring, introduced the concept of a quasi-progenerator to characterize the equivalences between a subcategory of right modules over one ring that is closed under submodules, epimorphic images, and direct sums and the category of all right modules over another ring. In the interim, employing the notion of linear compactness, B. J. Müller had characterized the reflexive modules under Morita duality and given a one-sided characterization of the bimodules inducing these dualities.