Definition:Radical of Ideal of Ring/Definition 1

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 $I$:

$\map \Rad I := \set {a \in A: \exists n \in \N_{>0} : a^n \in I}$

Also see

• Equivalence of Definitions of Radical of Ideal of Ring

Sources

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