Modules

**peteryellow** · February 5th 2009, 01:53 PM

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 .

**NonCommAlg** · February 5th 2009, 04:23 PM

Originally Posted by peteryellow

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

they're defined very naturally:

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

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