Commutative Algebra/Sequences of modules

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Modules in category theory[edit]

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[edit]

Theorem 10.?:

Let be a ring and let be multiplicatively closed. Let be -modules. Then

exact implies exact.

-category-theoretic comment