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Math Help - groups and rings

  1. #1
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    groups and rings

    Show that if f and g are homomorphisms from Q into a ring R, Q the field of rational numbers, then if f(m) = g(m) for every integer m, f=g.

    Any ideas?
    Last edited by mr fantastic; April 30th 2009 at 02:08 PM. Reason: Restored question and closed thread
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  2. #2
    Super Member Gamma's Avatar
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    Field of Fractions

    Note f(1)=1_R=g(1)
    \forall n,m \in \mathbb{Z}
    f(n)=f(n*1)=nf(1)=ng(1)=g(n)
    f(1/m)=f(m^{-1})=f(m)^{-1}=g(m)^{-1}=g(m^{-1})=g(1/m)
    Thus for any q/s \in \mathbb{Q}
    f(q/s)=f(q*(1/s))=f(q)f(1/s)=g(q)g(1/s)=g(q/s)
    QED
    Last edited by Gamma; April 28th 2009 at 11:31 AM. Reason: TeX error
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