nilradical Nilradical 2002-06-17 00:08:01 2002-06-17 00:08:01 Definition nilpotent % 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 Let $R$ be a commutative ring. An element $x \in R$ is said to be {\em nilpotent} if $x^n = 0$ for some positive integer $n$. The set of all nilpotent elements of $R$ is an ideal of $R$, called the {\em nilradical} of $R$ and denoted $\operatorname{Nil}(R)$. The nilradical is so named because it is the radical of the zero ideal. The nilradical of $R$ equals the prime radical of $R$, although proving that the two are equivalent requires the axiom of choice.