$(-a)^2=a^2$ in commutative ring? Maybe this is a silly question, but 

how can I show that $(-a)^2=a^2$ in a commutative ring with $1$ for all $a$ in the ring? 

I know that $(-a)^2=(-a)\cdot(-a) =(-1)\cdot(-1)\cdot a^2$. So I have to show that $(-1)^2=1$, in other words, I have to prove that the inverse of $-1$ is $-1$. And is it necessary to have a $1$?  Thanks.

Hint: Try expanding $(1 + (-1))^2$.
