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.