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Math Help - Homomorphisms and Factor Rings

  1. #1
    Super Member Bernhard's Avatar
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    Homomorphisms and Factor Rings

    Let R and R' be rings and let N and N' be ideals of R and R' respectively.

    Let  \phi be a homomorphism of R into R'.

    Show that  \phi induces a natural homomorphism  \phi_*  : R/N \rightarrow R'/N'      if   \  \ \phi [N] \subseteq N'
    Last edited by Bernhard; February 21st 2013 at 02:17 AM.
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  2. #2
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    Re: Homomorphisms and Factor Rings

    well, what do we have to work with?

    we are given the homomorphism φ:R-->R', the ideal N of R and the ideal N' of R', and that φ(N) is a subset of N'.

    so the natural thing to do is define: φ*(r+N) = φ(r)+N'.

    whenEVER you define things on cosets, it is imperative that you verify that the definition depends ONLY on the coset r+N, and not on "r".

    so we must check that if r'+N = r+N, that φ*(r+N) = φ*(r'+N).

    if r+N = r'+N, this means r-r' is in N. since φ maps N inside N', φ(r-r') is in N'. since φ is a homomorphism, φ(r-r') = φ(r)-φ(r').

    so we have φ(r)-φ(r') is in N', hence φ(r)+N' = φ(r')+N', that is: φ*(r+N) = φ*(r'+N), as desired.

    now all that is left to do is verify that φ* is a ring homomorphism. you can do this.
    Thanks from Bernhard
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  3. #3
    Super Member Bernhard's Avatar
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    Re: Homomorphisms and Factor Rings

    Thanks Deveno ... appreciate your help.

    WIll now work through the post

    Peter
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