$\newcommand{\ad}{\operatorname{ad}}$Let $R$ be an associative ring. Set $[x, y] = xy - yx$ and $\ad_x(-) = [x, -]: R \to R$.
- Is there a formula for $(ab)^n$ in general? I found one formula for the group theoretic commutator, i.e., $$ (xy)^n = x^{\binom{n}1} y^{\binom{n}1} [y,x]^{\binom{n}{2}}[[y,x],x]^{\binom{n}{3}} \cdots [\cdots[[[ y, x],x],x]\cdots,x]^{\binom{n}{n}}, $$ I am wondering if there is any nice formula for the ring theoretic commutator. BTW, I found https://mathoverflow.net/q/78813/19222, which is a for $(x+y)^n$.
- In characteristic $p$, suppose that $r$ commutes with $[x, r]$, we should have $\ad_{rx}^{p-1}(r) = -r\ad_x^{p-1}(r^{p-1})$. But now I have no idea how to prove it.
Any hints? Thank you!