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).