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Math Help - Homomorphism of rings, inverses map to inverse?

  1. #1
    Ant
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    Homomorphism of rings, inverses map to inverse?

    Hi,

    If \phi: R \to S is a homomorphism between two rings R and S.

    Is it true that:

    If  a \in R is a unit then \phi(a^{-1})=\phi(a)^{-1}, or more generally do units have to map to units?

    Thanks
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  2. #2
    Ant
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    Re: Homomorphism of rings, inverses map to inverse?

    Sorry, just got it:

    \phi(1)=1 \implies \phi(a \cdot a^{-1}) = \phi(a)\cdot \phi(a^{-1}) = 1 \implies \phi(a)^{-1} = \phi(a^{-1})
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