Commutative Algebra/Sequences of modules

Modules in category theory

Definition 10.1 (${\displaystyle R}$-mod):

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

We aim now to prove that if ${\displaystyle R}$ is a ring, ${\displaystyle R}$-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 ${\displaystyle R}$-modules ${\displaystyle M,N}$ and a morphism ${\displaystyle f:M\to N}$ we define

${\displaystyle \ker f:=\{m\in M|f(m)=0\}}$.

Sequences of augmented modules

Theorem 10.?:

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

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

-category-theoretic comment