# Radical of an ideal

From Commalg

## Definition

Let be a commutative unital ring and be an ideal in . The **radical** of , sometimes denoted , is defined in the following equivalent ways:

- It is the set of all for which some positive power of lies inside
- It is the smallest radical ideal containing
- It is the intersection of all prime ideals containing
- Under the quotient map , it is the inverse image of the nilradical of

is a radical ideal iff .

### Equivalence of definitions

After quotienting out, the equivalence of definitions follows from the equivalence of various definitions of the nilradical.