MultiplicativeFilters 2007-03-06 13:38:30 2007-03-06 14:29:46 Example filter Gabriel Filter Multiplicative Filter % 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 \newcommand\sep{\,:\,} For any ring $A$, any set $S\subset A$ and any element $x\in A$, we use the notation $$(S:x):=\{ a\in A\ ax\in S\}$$ Let $A$ be a commutative ring with unity, and let $\mathcal{I}(A)$ be the set of all ideals of $A$. \begin{itemize} \item A \emph{Multiplicative Filter} of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that $I,J\in\mathcal{F}\Rightarrow IJ\in\mathcal{F}$. \item A \emph{Gabriel Filter} of $A$ is a filter $\mathcal{F}$ on $\mathcal{I}(A)$ such that $$ [I\in\mathcal{F},J\in\mathcal{I}(A)\textrm{ and }\forall x\in I,(J:x)\in\mathcal{F}]\Rightarrow J\in\mathcal{F} $$ \end{itemize} Note that Gabriel Filters are also Multiplicative Filters.