| Version 10 |
Version 9 |
| The {\em Jacobson radical} $J(R)$ of a ring $R$ is the intersection |
The {\em Jacobson radical} $J(R)$ of a ring $R$ is the intersection |
| of the annihilators of irreducible left $R$-modules. |
of the annihilators of irreducible left $R$-modules. |
| The following are alternate characterizations of the Jacobson radical $J(R)$: |
The following are alternate characterizations of the Jacobson radical $J(R)$: |
| \begin{enumerate} |
\begin{enumerate} |
| \item The intersection of all left primitive ideals. |
\item The intersection of all left primitive ideals. |
| \item The intersection of all maximal left ideals. |
\item The intersection of all maximal left ideals. |
| \item The set of all $t \in R$ such that for all $r \in R$, $1-rt$ is |
\item 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 $u(1-rt)=1$). |
left invertible (i.e. there exists $u$ such that $u(1-rt)=1$). |
| \item The largest ideal $I$ such that for all $v \in I$, $1-v$ is a |
\item The largest ideal $I$ such that for all $v \in I$, $1-v$ is a |
| unit in $R$. |
unit in $R$. |
| \item (1) - (3) with "left" replaced by "right" and $rt$ replaced by $tr$. |
\item (1) - (3) with "left" replaced by "right" and $rt$ replaced by $tr$. |
| \end{enumerate} |
\end{enumerate} |
| Note that if $R$ is commutative and finitely generated, then $J(R)=\{x \in R |
Note that if $R$ is commutative and finitely generated, then $J(R)=\{x \in R |
|
: x^n=0 \text{for some } n \in \mathbb{N} \} = \operatorname{Nil}(R)$$.
|
: x^n=0 \text{for some } n \in \mathbb{N} \}$.
|
