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Math Help - proofs using the axioms of rings

  1. #1
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    proofs using the axioms of rings

    use the axioms for a ring to prove the following:

    for all x,y,z in a ring

    -1*x = -x

    I can pull through if I can use the fact that 1x = x ... although I am unsure if this is allowed or not. Any help is appreciated!
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by minivan15 View Post
    use the axioms for a ring to prove the following:

    for all x,y,z in a ring

    -1*x = -x

    I can pull through if I can use the fact that 1x = x ... although I am unsure if this is allowed or not. Any help is appreciated!
    Since \mathbf{0} exists in a ring, we see that

    \begin{aligned}\left(-1\right)x&=\left(-1\right)x+\mathbf{0}\\&=\left(-1\right)x+\left(x+\left(-x\right)\right)\\&=\left[\left(-1\right)x+x\right]+\left(-x\right)\\&=\left[\left(-1\right)+1\right]x+\left(-x\right)\\&=0x+\left(-x\right)\\&=\mathbf{0}+\left(-x\right)\\&=-x\qquad\blacksquare\end{aligned}

    Does this make sense?
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  3. #3
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    it does...although the axioms for a ring never state that 1x = x. Could you show me a proof of this? Other than that it is perfectly clear, thanks
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