Jacobson Radical - Properties

Properties

• If R is unital and is not the trivial ring {0}, the Jacobson radical is always distinct from R since rings with unity always have maximal right ideals. However, some important theorems and conjectures in ring theory consider the case when J(R) = R - "If R is a nil ring (that is, each of its elements is nilpotent), is the polynomial ring R equal to its Jacobson radical?" is equivalent to the open Köthe conjecture. (Smoktunowicz 2006, p. 260, §5)

• The Jacobson radical of the ring R/J(R) is zero. Rings with zero Jacobson radical are called semiprimitive rings.

• A ring is semisimple if and only if it is Artinian and its Jacobson radical is zero.

• 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's lemma).

• J(R) contains all central nilpotent elements, but contains no idempotent elements except for 0.

• J(R) contains every nil ideal of R. If R is left or right Artinian, then J(R) is a nilpotent ideal. This can actually be made stronger: If is a composition series for the right R-module R (such a series is sure to exist if R is right artinian, and there is a similar left composition series if R is left artinian), then . (Proof: Since the factors are simple right R-modules, right multiplication by any element of J(R) annihilates these factors. In other words, whence . Consequently, induction over i shows that all nonnegative integers i and u (for which the following makes sense) satisfy . Applying this to u = i = k yields the result.) Note, however, that in general the Jacobson radical need not consist of only the nilpotent elements of the ring.

• If R is commutative and finitely generated as a Z-module, then J(R) is equal to the nilradical of R.

• The Jacobson radical of a (unital) ring is its largest superfluous right (equivalently, left) ideal.