Morita duality

Summary

A Morita duality between two rings $R$ and $S$ is a contravariant equivalence between suitable categories of left $R$-modules and right $S$-modules. It is given by a bimodule $U$ through the functors $\Hom(-, U)$. It generalizes the duality $V \mapsto V^*$ between finite-dimensional vector spaces. Matlis duality over complete Noetherian local rings and the duality $\Hom(-, R)$ over quasi-Frobenius rings are the main examples. Morita characterized such dualities by the bimodule $U$, which must be an injective cogenerator on both sides.