quasi-regularity QuasiRegularity 2002-12-07 17:57:09 2006-12-11 12:50:34 Definition quasi-regular right quasi-regular left quasi-regular quasi-inverse quasi-regular ideal quasi-regular 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 \newtheorem*{lemma}{Lemma} \PMlinkescapeword{terms} An element $x$ of a ring is called \emph{right quasi-regular} [resp. \emph{left quasi-regular}] if there is an element $y$ in the ring such that $x + y + xy = 0$ [resp. $x + y + yx = 0$]. For calculations with quasi-regularity, it is useful to introduce the operation $*$ defined: $$ x * y = x + y + xy .$$ Thus $x$ is right quasi-regular if there is an element $y$ such that $x * y = 0$. The operation $*$ is easily demonstrated to be associative, and $x * 0 = 0 * x = x$ for all $x$. An element $x$ is called \emph{quasi-regular} if it is both left and right quasi-regular. In this case, there are elements $y$ and $z$ such that $x + y + xy = 0 = x + z + zx$ (equivalently, $x * y = z * x = 0$). A calculation shows that $$y = 0 * y = (z * x) * y = z * (x * y) = z.$$ So $y = z$ is a unique element, depending on $x$, called the \emph{quasi-inverse} of $x$. An ideal (one- or two-sided) of a ring is called \emph{quasi-regular} if each of its elements is quasi-regular. Similarly, a ring is called \emph{quasi-regular} if each of its elements is quasi-regular (such rings cannot have an identity element). \begin{lemma} Let $A$ be an ideal (one- or two-sided) in a ring $R$. If each element of $A$ is right quasi-regular, then $A$ is a quasi-regular ideal. \end{lemma} This lemma means that there is no extra generality gained in defining terms such as right quasi-regular left ideal, etc. Quasi-regularity is important because it provides elementary characterizations of the Jacobson radical for rings without an identity element: \begin{itemize} \item The Jacobson radical of a ring is the sum of all quasi-regular left (or right) ideals. \item The Jacobson radical of a ring is the largest quasi-regular ideal of the ring. \end{itemize} For rings with an identity element, note that $x$ is [right, left] quasi-regular if and only if $1 + x$ is [right, left] invertible in the ring.