Category of modules

Example

Let $R$ be a ring with identity. The category of left $R$-modules $R\text{-}\mathbf{Mod}$ has the left $R$-modules as objects and the $R$-linear maps as morphisms, with composition of maps. A map $f \colon M \to N$ is $R$-linear when $f(x + y) = f(x) + f(y)$ and $f(rx) = r f(x)$ for all $x, y \in M$ and $r \in R$. It is an abelian category. Each $\Hom_R(M, N)$ is an abelian group under pointwise addition, and composition is bilinear. The zero module is a zero object, and the direct sum $M \oplus N$ is both product and coproduct. The kernel of $f \colon M \to N$ is the submodule $\set{x \in M : f(x) = 0}$ and the cokernel is $N / f(M)$. As in abelian groups, each injective linear map is the kernel of its cokernel and each surjective linear map is the cokernel of its kernel.