Making an Idempotent Right Ideal

Question

Let $R$ be an associative ring with identity and let $a\in R$ and $x\in RaR$ (the ideal consisting of all finite sums of the form ${\sum^n_{i=1}}x_iay_i$) be such that $a-ax\in N^*(R)$, where $N^*(R)$ is the upper nil radical of $R$ which is the largest nil ideal of $R$. I think that the right ideal $axR$ is idempotent but I could not make through. Thanks for any help!

Answer 1

Take $R=\mathbb{Z}_8$ and $a=x=2$, then $a-ax=6\in N^*(R)=\{0,2,4,6\}$ and $axR=4\mathbb{Z}_8$ is not an idempotent ring.