Morita Theory for Rings

The basic philosophy of Morita theory is to study rings in terms of their modules. With this in mind, we say that two rings $R$ and $S$ are Morita equivalent if $Mod_R$ (the category of right $R$-modules) and $Mod_S$ are equivalent as categories. Properties preserved by Morita equivalence are essentially those that can be phrased in categorical language (that is to say, without reference to the elements of modules or of the ring itself). In this talk, I will discuss the framework for Morita theory (the so-called “Morita context”) and show that $R$ and $S$ are Morita equivalent if and only if $R$ is isomorphic to the endomorphism ring of some progenerator of $Mod_S$.