# Nilradical

From Commalg

*This article defines an ideal-defining function, viz a rule that inputs a commutative unital ring and outputs an ideal of that ring*

## Definition

### 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).