Definition for commutative rings
The Jacobson radical of a commutative unital ring is defined in the following equivalent ways:
- It is the intersection of all its maximal ideals
- It is the set of those elements which are in the kernel of every homomorphism to a field
- Nilradical: The nilradical is defined as the intersection of all prime ideals, and also as the set of all nilpotent elements. The nilradical is contained in the Jacobson radical. The two are equal for Jacobson rings (and for some other rings as well) but are not equal in general.
A key difference between the Jacobson radical and the nilradical is that there is in general no way to relate the Jacobson radical of a ring with that of a subring; however, the nilradical of a subring is precisely the intersection of the subring with the nilradical of the whole ring (this is clear from the characterization of the nilradical as the set of nilpotent elements). Thus, if both the ring and the subring are Jacobson rings, then the Jacobson radical of the subring equals the intersection of the subring with the Jacobson radical of the ring.