Commutative Algebra/Sequences of modules

Modules in category theory 0 100% #FFFFFF 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. Theorem 10.1: The category of modules has kernels. Proof: For -modules and a morphism we define . Sequences of augmented modules 0 100% #FFFFFF Theorem 10.?: Let be a ring and let be multiplicatively closed. Let be -modules. Then exact implies exact. -category-theoretic comment