Radical of an ideal

Definition

Let $R$ be a commutative unital ring and $I$ be an ideal in $R$. The radical of $I$, sometimes denoted $\sqrt{I}$, is defined in the following equivalent ways:

• It is the set of all $a\in R$ for which some positive power of $a$ lies inside $I$
• It is the smallest radical ideal containing $I$
• It is the intersection of all prime ideals containing $I$
• Under the quotient map $R\to R/I$, it is the inverse image of the nilradical of $R/I$

$I$ is a radical ideal iff $\sqrt{I}=I$.

Equivalence of definitions

After quotienting out, the equivalence of definitions follows from the equivalence of various definitions of the nilradical.