INTRODUCTION

The paper [Mo58] by Kiiti Morita seems to be the first systematic study of equivalences between module categories. Morita treats both contravariant equivalences (which he calls dualities of module categories) and covariant equivalences (which he calls isomorphisms of module categories) and shows that they always arise from suitable bimodules, either via contravariant hom functors (for ‘dualities’) or via covariant hom functors and tensor products (for ‘isomorphisms’). The term ‘Morita theory’ is now used for results concerning equivalences of various kinds of module categories. The authors of the obituary article [AGH] consider Morita's theorem “probably one of the most frequently used single results in modern algebra”.

In this survey article, we focus on the covariant form of Morita theory, so our basic question is:

When do two ‘rings’ have ‘equivalent’ module categories?

We discuss this question in different contexts:

• (Classical) When are the module categories of two rings equivalent as categories?

• (Derived) When are the derived categories of two rings equivalent as triangulated categories?

• (Homotopical) When are the module categories of two ring spectra Quillen equivalent as model categories?

There is always a related question, which is in a sense more general: What characterizes the category of modules over a ‘ring’?

The answer is, mutatis mutandis, always the same: modules over a ‘ring’ are characterized by the existence of a ‘small generator’, which plays the role of the free module of rank one.