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?