The $J(R)$ of a unital ring $R$ is the intersection of the annihilators of simple (http://planetmath.org/SimpleModule) left $R$-modules.

The following are alternative characterizations of the Jacobson radical $J(R)$:

1. 1.

The intersection of all left primitive ideals.

2. 2.

The intersection of all maximal left ideals.

3. 3.

The set of all $t\in R$ such that for all $r\in R$, $1-rt$ is left invertible (i.e. there exists $u$ such that $u(1-rt)=1$).

4. 4.

The largest ideal $I$ such that for all $v\in I$, $1-v$ is a unit in $R$.

5. 5.

(1) - (3) with “left” replaced by “right” and $rt$ replaced by $tr$.

If $R$ is commutative and finitely generated, then

 $J(R)=\{x\in R\mid x^{n}=0\hbox{ for some }n\in\mathbb{N}\}=\operatorname{Nil}(% R).$

The Jacobson radical can also be defined for non-unital rings. To do this, we first define a binary operation $\circ$ on the ring $R$ by $x\circ y=x+y-xy$ for all $x,y\in R$. Then $(R,\circ)$ is a monoid, and the Jacobson radical is defined to be the largest ideal $I$ of $R$ such that $(I,\circ)$ is a group. If $R$ is unital, this is equivalent to the definitions given earlier.

 Title Jacobson radical Canonical name JacobsonRadical Date of creation 2013-03-22 12:36:11 Last modified on 2013-03-22 12:36:11 Owner yark (2760) Last modified by yark (2760) Numerical id 19 Author yark (2760) Entry type Definition Classification msc 16N20 Related topic Annihilator Related topic RadicalOfAnIdeal Related topic SimpleModule Related topic Nilradical Related topic RadicalTheory Related topic QuasiRegularity