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