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## Homework Statement

Let R be a ring and suppose there exists a positive even integer n such that x^n = x for

every x in R. Show that -x = x for every x in R.

## Homework Equations

## The Attempt at a Solution

I solved the case where n = 2.

Let x be in R.

(x+x)^2= x+x = 2x,

(x+x)^2 = 4x^2 = 4x.

So 4x = 2x and 2x = 0. Done.

I tried using this same method when n = 4 and got nowhere.