Nilradical of ring R, $\text{nil}(R)$ is intersection of prime ideals, is radical of $(0)$.
So I feel the name 'radical' is natural.
But Jacobson radical, that is an intersection of maximal ideals, seem not to be radical of a certain ideal.
Why Jacobson radical is nevertheless called 'radical'? Thank you.
Best Answer
The term "radical" here doesn't refer to the radical of some ideal, but to the general concept of a "radical of a ring" which is in some way "a set of bad elements". For nilradical, "bad" means nilpotent; for Jacobson radical, "bad" means annihilating all simple left modules.