nil and nilpotent ideals NilAndNilpotentIdeals 2002-12-08 18:48:35 2003-09-04 20:04:30 Definition nil nil ring nil ideal nil right ideal nil left ideal nil subring nilpotent nilpotent element nilpotent ring nilpotent ideal nilpotent right ideal nilpotent left ideal nilpotent subring locally nilpotent locally nilpotent ring locally nilpotent ideal locally nilpotent right ideal locally nilpotent left ideal locally nilpotent subring % 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 An element $x$ of a ring is \emph{nilpotent} if $x^n = 0$ for some positive integer $n$. A ring $R$ is \emph{nil} if every element in $R$ is nilpotent. Similarly, a one- or two-sided ideal is called \emph{nil} if each of its elements is nilpotent. A ring $R$ [resp. a one- or two sided ideal $A$] is \emph{nilpotent} if $R^n = 0$ [resp. $A^n = 0$] for some positive integer $n$. A ring or an ideal is \emph{locally nilpotent} if every finitely generated subring is nilpotent. The following implications hold for rings (or ideals): $$\text{nilpotent} \quad\Rightarrow \text{locally nilpotent} \quad\Rightarrow \text{nil}$$