Jacobson radical

Given a unital (typically noncommutative) ring $R$, the **Jacobson radical** $J(R)$ is defined as the set of elements $r\in R$ satisfying the following equivalent properties:

- For each simple left $R$-module $M$, $r M =0$.
- Each maximal left ideal of $R$ contains $r$.
- For all $x\in R$, $1 - r x$ is left invertible in $R$.

Alternatively,

- $J(R)$ is the intersection of all maximal left ideals of $R$.
- $J(R)$ is the intersection of all maxaimal right ideals of $R$.

The properties required remain the same if one interchanges left and right (modules, invertibility etc.) i.e. $J(R)=J(R^{op})$.

$J(R)$ is a $2$-sided ideal in $R$. The rings for which $J(R)=0$ are called **semiprimitive rings**. In other words, for each nonzero element $r$ in a semiprimitive ring, by the definition, there is a simple module left annihilated by $r$. Given any ring $R$, the quotient $R/J(R)$ is semiprimitive.

Revised on February 19, 2010 16:34:31
by Zoran Škoda
(161.53.130.104)