$N(R)$ when $R$ is a P.I. ring

Question

The set of nilpotent elements $N(R)$ of a ring $R$ with identity is not necessarily a right ideal (or even a subgroup) as it is seen in the ring of $2×2$ matrices over $\mathbb Z$. But, my question concerns PI-rings (polynomial identity rings). Is it true that $N(R)$ is a right (or left) ideal of $R$? The answer is in affirmative when $R$ is commutative (which is a special case of a PI-ring).

Thanks for any suggestion!

Answer 1

You almost answer your own question, since $M_2(\mathbb{Z})$ is a PI-ring.

See https://en.wikipedia.org/wiki/Amitsur%E2%80%93Levitzki_theorem