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