All rings below are assumed to be commutative with unity.
It is well-known that in a Noetherian ring, the nilradical (the ideal of all nilpotent elements , or equivalently, the intersection of all prime ideals) is nilpotent (https://en.wikipedia.org/wiki/Nilpotent_ideal ) .
How much (or even at all) can we go in the converse direction ?
For a more explicit question, can we characterize those rings which satisfy a.c.c. on radical ideals and also the nilradical is nilpotent ?
(the only thing I am able to see is that in a ring satisfying a.c.c. on radical ideals, radical of every ideal is a finite intersection of prime ideals, hence the nilradical also is a finite intersection of prime ideals. )