Let R be a ring such that $a^2 = a$ , $\forall a$ $\in R$ . Prove that R is commutative. Im not sure what to do for this one, any help would be appreciated. 

$$(a+b)^2 = (a+b) \Rightarrow a^2+ab+ba+b^2 = a+b$$ Since $a^2 = a$ and $b^2 = b$ we have that $ab = -ba$ by cancellation. Now $$(a+a) = (a+a)^2 = a^2+2a^2+a^2= a+a+a+a \Rightarrow a+a = 0 \text{ for all } a\in \mathbb{R}.$$ This says that each element is its own (additive) inverse and so $-ba = ba$ which finally gives $$ab = -ba = ba$$
