Jacobson radical JacobsonRadical 2002-04-20 02:26:34 2008-01-03 10:50:59 Definition % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here The {\em Jacobson radical} $J(R)$ of a unital ring $R$ is the intersection of the annihilators of \PMlinkname{simple}{SimpleModule} left $R$-modules. The following are alternative characterizations of the Jacobson radical $J(R)$: \begin{enumerate} \item The intersection of all left primitive 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 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 unit in $R$. \item (1) - (3) with ``left'' replaced by ``right'' and $rt$ replaced by $tr$. \end{enumerate} 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.