Radical of an ideal
- It is the set of all for which some positive power of lies inside
- It is the smallest radical ideal containing
- It is the intersection of all prime ideals containing
- Under the quotient map , it is the inverse image of the nilradical of
is a radical ideal iff .
Equivalence of definitions
After quotienting out, the equivalence of definitions follows from the equivalence of various definitions of the nilradical.