radical of an ideal

Definition

Let be a commutative ring and an ideal. The radical of (denoted ) is

An ideal is called radical if for an element , then for some integer , that is

In words, if a higher power is in the ideal, then the element itself is in the ideal.

Alternate definition using prime ideals

Let be an ideal, then

Relationship to the quotient

If is radical, then the quotient is reduced.