In any ring R, show that ab is nilpotent iff ba is nilpotent. Can one say the same for zero divisors?

Question

I tried it this way... $(ab)^{n}=o$ then $a(ba)^{n-1}b=0$ but then we can't conclude anything from this as a and b may not have inverses..

Thanks for any help!!

Answer 1

Try it the other way around: If $(ab)^n = 0$ then $b(ab)^na = 0$.

Answer 2

$(ab)^n=0$ implies $b(ab)^na=(ba)^{n+1}=0$.