PrimaryIdeal 2004-03-12 12:57:12 2008-10-05 11:28:04 Definition primary $P$-primary % 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} \usepackage{amsthm} % 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 \newcommand{\mc}{\mathcal} \newcommand{\mb}{\mathbb} \newcommand{\mf}{\mathfrak} \newcommand{\ol}{\overline} \newcommand{\ra}{\rightarrow} \newcommand{\la}{\leftarrow} \newcommand{\La}{\Leftarrow} \newcommand{\Ra}{\Rightarrow} \newcommand{\nor}{\vartriangleleft} \newcommand{\Gal}{\text{Gal}} \newcommand{\GL}{\text{GL}} \newcommand{\Z}{\mb{Z}} \newcommand{\R}{\mb{R}} \newcommand{\Q}{\mb{Q}} \newcommand{\C}{\mb{C}} \newcommand{\<}{\langle} \renewcommand{\>}{\rangle} An ideal $Q$ in a commutative ring $R$ is a \emph{primary ideal} if for all elements $x,y\in R$, we have that if $xy\in Q$, then either $x\in Q$ or $y^n\in Q$ for some $n\in\mb{N}$. This is clearly a generalization of the notion of a prime ideal, and (very) loosely mirrors the relationship in $\mb{Z}$ between prime numbers and prime powers. It is clear that every prime ideal is primary. \textbf{Example.} Let $Q=(25)$ in $R=\mb{Z}$. Suppose that $xy\in Q$ but $x\notin Q$. Then $25|xy$, but 25 does not divide $x$. Thus 5 must divide $y$, and thus some power of $y$ (namely, $y^2$), must be in $Q$. The radical of a primary ideal is always a prime ideal. If $P$ is the radical of the primary ideal $Q$, we say that $Q$ is \emph{$P$-primary}. %If the radical of the primary ideal $Q$ is the prime ideal $P$, then $Q$ is said to be \emph{$P$-primary}.