Proving commutativity

Question

Let $R$ be a ring in which $x^2=x$ for all $x\in R$ where $x^2$ of course denotes $x\cdot x$.

• a. prove that $x+x=0$, for all $x \in R$
• b. prove that $R$ is commutative.

I have done part a but how do you do part b?

Answer 1

Hint: Insert $x=a+b$ into the identity $x^2=x$.

Answer 2

You want to prove $xy=yx$ for given elements $x,y \in X$. This is equivalent to $xy+yx=0$ by a.

There is nothing we can exploit from $x^2=x$ and $y^2=y$. We have to take some element which relates $x$ and $y$. So what about $x+y$?

We have $(x+y)^2=x+y$. Now expand the left hand side, cancel $x$ and $y$, and you are done.