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Math Help - Modules

  1. #1
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    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 .
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  2. #2
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    Quote Originally Posted by peteryellow View Post

    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.
    Last edited by NonCommAlg; February 5th 2009 at 03:53 PM.
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