TITLE: Prove that $R$ is commutative
QUESTION [5 upvotes]: Let $R$ be a ring such that
  $$ x^2-x \in C(R) \quad \forall x \in R,$$
  where $C(R)$ is the center of $R$. Prove that $R$ is commutative.

I cannot find any route to proceed. How can I start? Please give hints instead of full solution.

REPLY [6 votes]: As a starting point, consider $(x+y)^2-(x+y)=(x^2-x)+(y^2-y)+xy+yx$.  Since the center of a ring is closed under addition, this implies that $xy+yx\in C(R)$.  Now, there are two cases: either there are no nonzero elements in the center, or there are.