Super Morita Equivalence

The theory of Morita equivalences of module categories is a key tool in many areas of algebra; the simplest and most prototypical Morita equivalence is the equivalence between the category of R-modules, for R an associative ring, and the category of Mn(R)-modules, where Mn(R) is the ring of n × n matrices with entries in R. We develop a generalization of the theory of Morita equivalences to categories of modules over super (that is, Z_2 -graded) rings, and apply our results to the context of super Azumaya algebras, where the equivalence can be made quite explicit.