Why Jacobson radical is 'radical'? 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.
 A: To add another viewpoint: the "bad element" description of radicals is somewhat useful, but I also feel like it might be someone's atttempt at a retrospective description that just caught on.
I tried tracking down the original usage in ring-theory prehistory just now, but I didn't get very far. A biographical article on Wedderburn hints that it may lie in the work of Cartan, Molien and Frobenius.  (It would be interesting to know where to find that but I reached the end of my timebox on the subject. )
Really you should probably interpret "radical" as a cousin of "$\sqrt{\cdot}$" descended from the same latin root meaning... "root."
A little caveat: The following explanation is speculative. I don't know if this was the original intention or if I am just the next person with a clever retrospective explanation.
The following description applies pretty well:

A root is something important that supports a structure but is itself largely hidden.  When viewing the elements through a certain lens, the "radical" for that lens is the set of elements which you can't see through the lens.

When your lens is representation theory, the elements that are invisible to the semisimple representations are the Jacobson radical elements. They don't act at all on any semisimple representations, so they are "hidden."
When your lens is classical algebraic geometry, the nilpotent elements are invisible to the Zariski topology.
I think this is essentially what is captured by the definition of a radical class and that the corresponding ideal is essentially what is invisible to the respective torsion theory.  I am not an expert in this subject, though, so take it with a grain of salt.
The elements themselves are not always obviously "bad" in the sense of "badly behaved." Are the elements of the maximal ideal of a local domain really 'bad'?  Not with respect to the ring operation, I don't think. But in terms of shedding light on semisimple modules for the ring, they don't contribute.
A: 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.
