proofs using the axioms of rings

• Apr 13th 2009, 07:37 PM
minivan15
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!
• Apr 13th 2009, 07:47 PM
Chris L T521
Quote:

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 $\displaystyle \mathbf{0}$ exists in a ring, we see that

\displaystyle \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?
• Apr 13th 2009, 09:18 PM
minivan15
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