1. The problem statement, all variables and given/known data Let r be an element of an integral domain R such that r^2 = r. Show that either r = 0_R or 1_R 2. Relevant equations integral domain means no zero divisors. 3. The attempt at a solution This is fundamental as 0 and 1 solve r^2 = r and are the only solutions. However, I'm not really sure what I can play with to show this fact. We have r*r = r There are no zero divisors so no such thing as r*s = 0. If we start this proof off by assuming r is not 1_R or 0_R then maybe we could get somewhere, but It doesn't feel promising. There has to be a way to use the fact that r^2 = r. (r^2)*r = r^2 ? Then we are left with r*r = r, that's no good. Is there any trick that I am neglecting? I feel this is just a simple multiplication of two things and boom we have r(r^2) = 0 or 1 or something. How about r^2 - r = r-r then we have r(r-1)=0 r is either 0 or 1. BOOM?