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Math Help - Homomorphism Quesion

  1. #1
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    Homomorphism Quesion

    If we have a ring homomorphism f:R->S, I know that

    f(x+y) = f(x) + f(y)
    f(xy) = f(x)f(y)

    But is it also true that f(x) - f(y) = f(x-y)?
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  2. #2
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    Re: Homomorphism Quesion

    let's see if we can prove it:

    f(x-y) = f(x+(-y)) = f(x)+f(-y).

    does f(-y) = -f(y)? well, let's see if we can prove it:

    f(y) + f(-y) = f(y+(-y)) = f(0) = 0, so f(-y) must be the additive inverse of f(y), -f(y).

    therefore (going back to our first proof, now) f(x-y) = f(x) + f(-y) = f(x) + -f(y) = f(x) - f(y). yep, it must be true.
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  3. #3
    MHF Contributor Drexel28's Avatar
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    Re: Homomorphism Quesion

    Going back to what was said earlier, I be the OP has already seen a fair amount of group theory, from where he should already know this since every ring homomorphism is a group homomorphism of the rings underlying abelian group.
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