Assume that $\forall a\in R, a^2+a\in \operatorname{cent}R$. Need to show that $R$ is a commutative ring.

Question

Let $R$ be a ring. Assume that $\forall a\in R, a^2+a\in \operatorname{cent}R$. I need to show that $R$ is a commutative ring. The author gives a hint; that is to show that $\forall a,b\in R, ab+ba\in \operatorname{cent}R$, and I did show that it happens. But I don't know how to proceed. Could someone please give me a second hint? Thank you.

Answer 1

Once you have $ab+ba$ in the centre, you have $a^2b+aba=aba+ba^2$ so that $a^2b=ba^2$. So $a^2$ commutes with $b$, but $a^2+a$ is in the centre....