Computation in a ring

Question

I'm not sure about whether in a ring $R$ with unity we have $(-1_R)^2=1_R$ and thus any even power of $-1_R$ is $1_R$.

Are my calculations correct ?

\begin{split} (-1_R)^2 = (-1)_R.\times(-1_R) \\ = -(1_R.(-1_R))\\ =-(-1_R)\\ =1_R \end{split}

Thanks for your concern.

Answer 1

In a ring, there are more general rules such as $(-a)(-b) = -(-a)b = -(a(-b)) = -(-(ab))$ and $-(-a) = a$.