# Definition:Radical of Ideal of Ring/Definition 1

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## Definition

Let $A$ be a commutative ring with unity.

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

The **radical of $I$** is the ideal of elements of which some power is in $I$:

- $\map {\operatorname {Rad} } I := \set {a \in A: \exists n \in \N : a^n \in I}$

## Also see

## Sources

- 1969: M.F. Atiyah and I.G. MacDonald:
*Introduction to Commutative Algebra*: Chapter $1$: Rings and Ideals: $\S$ Operations on Ideals