Prove: if x^n = x, then prove that x + x = 0 for additive identity 0. Suppose $n$ is a positive integer.
Under a ring, suppose for any $x$, $x^n = x$. Then for any $x$, $x + x = 0$ for additive identity $0$. 
How is this proven?
The case when $n$ is even is rather easy: 
$x^n = (-x)^n$ so $x = -x$
However I am not able to find a proof for a general $n$.

The statement is false. The In $\mathbb Z_3$, you have $2^3=2$ and $2+2=1\neq 0$.
