the sum of the invertible elements of a finite ring:$S^2=S$ or $S^2=0$.

Question

Let S be the sum of the invertible elements of a finite ring. Prove that $S^2=S$ or $S^2=0$. If $1+1 \ne 0$ then $x \ne -x$ so $S=0$. I do not know how to show it when $1+1=0$. Can somebody help me, please?

Answer 1

Hint: Take the list of all $k$ invertible elements. What happens when we multiply all of them (pick a side, but make sure it's the same side every time) by some invertible element $x$?