Let $R$ be a ring, and let $C$={$x\in R: xy=yx$ for all y in $R$}

Question

Let $R$ be a ring, and let $C$={$x\in R: xy=yx$ for all y in $R$}

Prove that if $x^2-x \in C$ for all $x$ in $R$, then $R$ is commutative.

There's a hint given in the book that says "Show that $xy+yx \in C$ by considering $x + y$, and then show that $x^2 \in C$".

I manage to show $xy+yx \in C$, but don't really know how to show $x^2 \in C$.

Any help is appreciated, thanks!

Answer 1

Hint:

$$ x^2y - yx^2 = x(xy + yx) - (yx + xy)x. $$