Modules in category theory
Definition 10.1 (-mod):
For each ring , there exists one category of modules, namely the modules over with module homomorphisms as the morphisms. This category is called -mod.
We aim now to prove that if is a ring, -mod is an Abelian category. We do so by verifying that modules have all the properties required for being an Abelian category.
The category of modules has kernels.
For -modules and a morphism we define
Sequences of augmented modules
Let be a ring and let be multiplicatively closed. Let be -modules. Then
- exact implies exact.