Nilradical of A Ring - Commutative Rings

Commutative Rings

The nilradical of a commutative ring is the set of all nilpotent elements in the ring, or equivalently the radical of the zero ideal. This is an ideal because the sum of any two nilpotent elements is nilpotent, and the product of any element with a nilpotent element is nilpotent. It can also be characterized as the intersection of all the prime ideals of the ring.

If the ring is artinian, the nilradical coincides with the Jacobson radical, and it is the maximal nilpotent ideal of the ring. In general, if the nilradical is finitely generated (e.g., the ring is noetherian), then it is nilpotent.