Definition:Radical of Ideal of Ring/Definition 2

Definition

Let $A$ be a commutative ring with unity.

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

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 see

• Equivalence of Definitions of Radical of Ideal of Ring

Sources

• 1972: N. Bourbaki: Commutative Algebra: Chapter $\text {II}$: Localization: $\S 2$: Rings and modules of fractions: $2.6$: Nilradical and minimal prime ideals: Definition $4$