Unless R is the trivial ring {0}, the Jacobson radical is always a proper ideal in R.
If R is commutative and finitely generated, then J(R) is equal to the nilradical of R.
The Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitive.
If f : R -> S is a surjective ring homomorphism, then f(J(R)) ⊆ J(S).
If M is a finitely generated left R-module with J(R)M = M, then M = 0 (Nakayama lemma).
J(R) contains every nil ideal of R. If R is left or right artinian, then J(R) is a nilpotent ideal. Note however that in general the Jacobson radical need not contain every nilpotent element of the ring.
See also: radical of a module.
Examples:
Properties
This article (or an earlier version of it) was based on the Jacobson radical article from PlanetMath.