Commutative Algebra/Sequences of modules

Modules in category theory

Definition 10.1 ($R$-mod):

For each ring $R$, there exists one category of modules, namely the modules over $R$ with module homomorphisms as the morphisms. This category is called $R$-mod.

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

Theorem 10.?:

Let $R$ be a ring and let $S\subseteq M$ be multiplicatively closed. Let $M,N,K$ be $R$-modules. Then

$0\rightarrow M\rightarrow N\rightarrow K\rightarrow 0$ exact implies $0\rightarrow S^{-1}M\rightarrow S^{-1}N\rightarrow S^{-1}K\rightarrow 0$ exact.

-category-theoretic comment