# Definition:Radical of Ideal of Ring

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## Definition

Let $A$ be a commutative ring with unity.

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

### Definition 1

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

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

### Definition 2

Let $A / I$ be the quotient ring of $A$ by $I$.

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

Let $\pi: A \to A / I$ be the quotient epimorphism from $A$ onto $A / I$.

The **radical** of $I$ is the preimage of $\Nil {A / I}$ under $\pi$:

- $\map \Rad I = \pi^{-1} \sqbrk {\Nil {A / I} }$

## Also denoted as

The **radical** of $I$ is also denoted as $\sqrt I$ or $\map r I$.

The notation $\sqrt I$ is not recommended, as it is to easy to conflate with the sign for the square root, and so will not be used on $\mathsf{Pr} \infty \mathsf{fWiki}$.

## Also see

- Equivalence of Definitions of Radical of Ideal of Ring
- Radical of Ideal of Ring is Ideal
- 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