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.