AntiIsomorphism 2006-06-17 23:09:39 2007-06-01 09:51:10 Definition anti-endomorphism anti-homomorphism anti-isomorphic anti-automorphism % 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 $R$ and $S$ be rings and $f: R\longrightarrow S$ be a function such that $f(r_{1}r_{2}) = f(r_{2})f(r_{1})$ for all $r_{1}, r_{2} \in R$. If $f$ is a homomorphism of the additive groups of $R$ and $S$, then $f$ is called an {\it anti-homomorphsim}. If $f$ is a bijection and anti-homomorphism, then $f$ is called an {\it anti-isomorphism}. If $f$ is an anti-homomorphism and $R=S$ then $f$ is called an {\it anti-endomorphism}. If $f$ is an anti-isomorphism and $R=S$ then $f$ is called an {\it anti-automorphism}. As an example, when $m \neq n$, the mapping that sends a matrix to its transpose (or to its conjugate transpose if the matrix is complex) is an anti-isomorphism of $M_{m,n} \to M_{n,m}$. $R$ and $S$ are \emph{anti-isomorphic} if there is an anti-isomorphism $R \to S$. All of the things defined in this entry are also defined for groups.