Jacobson radical
The Jacobson radical^{} $J(R)$ 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.
The intersection of all left primitive ideals.

2.
The intersection of all maximal left ideals^{}.

3.
The set of all $t\in R$ such that for all $r\in R$, $1rt$ is left invertible (i.e. there exists $u$ such that $u(1rt)=1$).

4.
The largest ideal $I$ such that for all $v\in I$, $1v$ is a unit in $R$.

5.
(1)  (3) with “left” replaced by “right” and $rt$ replaced by $tr$.
If $R$ is commutative^{} and finitely generated^{}, then
$$J(R)=\{x\in R\mid {x}^{n}=0\text{for some}n\in \mathbb{N}\}=\mathrm{Nil}(R).$$ 
The Jacobson radical can also be defined for nonunital rings. To do this, we first define a binary operation^{} $\circ $ on the ring $R$ by $x\circ y=x+yxy$ 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  20130322 12:36:11 
Last modified on  20130322 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 