Definition:Radical of Ideal of Ring

Definition
Let $A$ be a commutative ring with unity.

Let $I$ be an ideal of $A$.

Definition 1
The radical of $I$, usually written $\operatorname{Rad} \left({I}\right)$ or $\sqrt I$, is defined as:


 * $\operatorname{Rad} \left({I}\right) := \left\{{a \in A: a^n \in I \text{ for some positive integer } n}\right\}$

Definition 2
Let $A/I$ be the quotient ring.

Let $\operatorname{Nil}(A/I)$ be its nilradical.

Let $\pi : A \to A/I$ be the quotient mapping.

The radical of $I$ is the preimage of $\operatorname{Nil}(A/I)$ under $\pi$:
 * $\operatorname{rad}(I) = \pi^{-1}(\operatorname{Nil}(A/I))$

Also see

 * Equivalence of Definitions of Radical of Ideal of Ring
 * Definition:Radical Ideal of Ring
 * Radical of Ideal is Intersection of Containing Prime Ideals

Special cases

 * Definition:Radical of Integer
 * Definition:Nilradical of Ring, the radical of the zero ideal

Generalizations

 * Definition:Radical of Subset of Ring