Nilradical

Symbol-free definition
The nilradical of a commutative unital ring is defined as the subset that satisfies the following equivalent conditions:


 * It is the intersection of all prime ideals
 * It is the intersection of all radical ideals
 * It is the radical of zero.
 * It is the set of nilpotent elements

Equivalence of definitions
For a proof of the equivalence of definitions, see nilradical is smallest radical ideal and nilradical equals intersection of all prime ideals (the remaining equivalences are direct from definitions).