# Thread: proofs using the axioms of rings

1. ## 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!

2. Originally Posted by minivan15
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?

3. 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