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 $r\in R$, $1-rt$ is left invertible (i.e. there exists $u$ such that $u(1-rt)=1$).

4. The largest ideal $I$ such that for all $v\in I$, $1-v$ 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 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