Thread: Modules

For any R–homomorphism ϕ: M → N
we define its cokernel by Coker ϕ = N/ Imϕ. Establish an exact sequence:
O → Ker ϕ −a→ M --ϕ−→ N--β−→ Coker ϕ → O .

Originally Posted by peteryellow

For any R–homomorphism $\displaystyle \phi: M \longrightarrow N.$ we define its cokernel by $\displaystyle \text{coker} \ \phi = N/ \text{im} \ \phi.$ Establish an exact sequence: $\displaystyle 0 \longrightarrow \ker \phi \overset{\alpha}{\longrightarrow} M \overset{\phi}{\longrightarrow} N \overset{\beta}{\longrightarrow} \text{coker} \ \phi \longrightarrow 0 $ .

they're defined very naturally:

$\displaystyle \alpha(x)=x$ and $\displaystyle \beta(y)=y + \text{im} \ \phi,$ for all $\displaystyle x \in \ker \phi$ and $\displaystyle y \in N.$ see that $\displaystyle \alpha$ is an injective and $\displaystyle \beta$ is a surjective homomorphism. also $\displaystyle \text{im} \ \alpha = \ker \phi,$ and $\displaystyle \ker \beta = \text{im} \ \phi.$ so the sequence is exact.