Radicals, Lower Nil Radical
Lower Nil Radical
Recall that the upper nil radical of a ring R is the largest nil ideal of R, written upnil(R).
The lower nil radical is the intersection of all prime ideals, written lownil(R).
This is a radical ideal, specifically, rad(0).
It is also a semiprime ideal,
the intersection of all the semiprime ideals of R,
and the smallest semiprime ideal in R.
Note that R/lownil(R) is a semiprime ring, since 0 is a semiprime ideal in the image.
If x lies in lownil(R) then the powers of x form an n-system that intersects 0.
(Otherwise 0 could rise to a semiprime ideal missing the powers of x.)
Thus x is nilpotent, and lownil(R) is a nil ideal.
This is contained in upnil(R), the largest nil ideal,
which is contained in jac(R) (holding all the nil ideals).
If R has no nil ideals, as when R is reduced
or jacobson semisimple, upnil(R) = lownil(R) = 0.
Commutative Ring
If R is commutative, the nilpotent elements form a nil ideal, which becomes the largest nil ideal of R.
This in turn is rad(0).
Thus lownil(R) = upnil(R), and we simply write nil(R).
Such a ring is reduced iff nil(R) = 0, iff R is a semiprime ring.
Left Artinian
Let R be left artinian, whence jac(R) is nilpotent.
Thus the jacobson radical lies in the semiprime ideal lownil(R),
and jac(R) = upnil(R) = lownil(R).