The Descent Theory of Rings

A morphism σ: R → of rings induces a pair of adjoint functors between the categories of R-modules and S-modules. Via this adjunction, the category of S-modules is always monadic over the category of R-modules: this implies that we can view an S-module as being an R-module with an additional structure. The morphism σ: R → S of rings is a morphism of effective descent when, moreover, the category of R-modules is co-monadic over the category of S-modules; in that case, each R-module can thus also be seen as an S-module with an additional structure. We prove that the effective descent morphisms of rings are exactly the pure ones: the injective morphisms, which remain injective when tensored with whatever R-module. The descent theorem for rings implies an analogous result for algebras.