The sum of nil one-sided ideals of bounded index of a ring

The sumN(R) of nil one-sided ideals of bounded index of a ringR is shown to coincide with the set of all strongly nilpotent elements ofR of bounded index. The known result thatN(R) is contained in the prime radical is highly improved and it is shownN(R) is contained inN ₂(R). It is proved that the sum of a finite number of nil left ideals of bounded index has bounded index.