topological ring TopologicalRing 2002-06-08 22:38:50 2008-06-28 02:46:01 Definition topological field topological division ring % 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 A ring $R$ which is a topological space is called a \emph{topological ring} if the addition, multiplication, and the additive inverse functions are continuous functions from $R \times R$ to $R$. A \emph{topological division ring} is a topological ring such that the multiplicative inverse function is continuous away from $0$. A \emph{topological field} is a topological division ring that is a field. \textbf{Remark}. It is easy to see that if $R$ contains the multiplicative identity $1$, then $R$ is a topological ring iff addition and multiplication are continuous. This is true because the additive inverse of an element can be written as the product of the element and $-1$. However, if $R$ does not contain $1$, it is necessary to impose the continuity condition on the additive inverse operation.