classical ring of quotients ClassicalRingOfQuotients 2003-10-20 20:39:12 2003-11-24 10:11:21 Definition regular % 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 ring. An element of $R$ is called \emph{regular} if it is not a right zero divisor or a left zero divisor in $R$. A ring $Q \supset R$ is a \emph{left classical ring of quotients} for $R$ (resp. \emph{right classical ring of quotients} for $R$) if it satisifies: \begin{itemize} \item every regular element of $R$ is invertible in $Q$ \item every element of $Q$ can be written in the form $x^{-1}y$ (resp. $yx^{-1}$) with $x, y \in R$ and $x$ regular. \end{itemize} If a ring $R$ has a left or right classical ring of quotients, then it is unique up to isomorphism. If $R$ is a commutative integral domain, then the left and right classical rings of quotients always exist -- they are the field of fractions of $R$. For non-commutative rings, necessary and sufficient conditions are given by Ore's Theorem. Note that the goal here is to construct a ring which is not too different from $R$, but in which more elements are invertible. The first condition says which elements we want to be invertible. The second condition says that $Q$ should \PMlinkescapetext{contain} just enough extra elements to make the regular elements invertible. Such rings are called classical rings of quotients, because there are other rings of quotients. These all attempt to enlarge $R$ somehow to make more elements invertible (or sometimes to make ideals invertible). Finally, note that a ring of quotients is not the same as a quotient ring.