**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.

Read more about this topic: Jacobson Radical

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