Nilradical

Definition for commutative rings

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

Definition for noncommutative rings

For noncommutative rings, there is no single nilradical, rather there is a general property of being a nilradical. Further information: Nilradical (noncommutative rings)