IdealOfAnAlgebra 2008-06-15 16:41:37 2008-06-18 00:36:59 Definition % 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 \PMlinkescapephrase{left ideal} \PMlinkescapephrase{right ideal} \PMlinkescapephrase{two-sided ideal} Let $A$ be an algebra over a ring $R$. {\bf Definition -} A \emph{left ideal} of $A$ is a subalgebra $I \subseteq A$ such that $ax \in I$ whenever $a \in A$ and $ x \in I$. Equivalently, a left ideal of $A$ is a subset $I \subset A$ such that \begin{enumerate} \item $x - y \in I$, for all $x, y \in I$. \item $rx \in I$, for all $r \in R$ and $x \in I$. \item $ax \in I$, for all $a \in A$ and $x \in I$ \end{enumerate} Similarly one can define a \emph{right ideal} by replacing condition 3 by: $xa \in I$ whenever $a \in A$ and $x \in I$. A \emph{two-sided ideal} of $A$ is a left ideal which is also a right ideal. Usually the word "\PMlinkescapetext{ideal}" by itself means two-sided ideal. Of course, all these notions coincide when $A$ is commutative. \subsubsection{Remark} Since an algebra is also a ring, one might think of borrowing the definition of ideal from ring \PMlinkescapetext{theory}. The problem is that condition 2 would not be in general satisfied (unless the algebra is unital).