We study the relationships between strongly prime ideals and completely prime ideals, concentrating on the connections among various radicals(prime radical, upper nilradical and generalized nilradical). Given a ring R, consider the condition: (*) nilpotent elements of R form an ideal in R. We show that a ring R satisfies (*) if and only if every minimal strongly prime ideal of R is completely prime if and only if the upper nilradical coincides with the generalized nilradical in R.