Jacobson radical

The Jacobson radical $J(R)$ of a unital ring $R$ is the intersection of the annihilators of simple 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$$ J(R)=\{x \in R \mid x^n=0 \hbox{ for some } n \in \mathbb{N} \} = \operatorname{Nil}(R).$$

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.