Commutative Algebra/Sequences of modules

Modules in category theory
We aim now to prove that if $$R$$ is a ring, $$R$$-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 $$R$$-modules $$M, N$$ and a morphism $$f: M \to N$$ we define
 * $$\ker f := \{m \in M|f(m) = 0\}$$.

Sequences of augmented modules
-category-theoretic comment