Characteristic2 2003-03-11 02:59:03 2007-05-31 02:47:54 Theorem cyclic 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{amsthm} \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 {\bf \PMlinkescapetext{Lemma}.} A finite ring is cyclic if and only if its \PMlinkname{order}{OrderRing} and characteristic are equal. \begin{proof} If $R$ is a cyclic ring and $r$ is a \PMlinkname{generator}{Generator} of the additive group of $R$, then $|r|=|R|$. Since, for every $s \in R$, $|s|$ divides $|R|$, then it follows that $\operatorname{char}~R=|R|$. Conversely, if $R$ is a finite ring such that $\operatorname{char}~R=|R|$, then the exponent of the additive group of $R$ is also equal to $|R|$. Thus, there exists $t \in R$ such that $|t|=|R|$. Since $\langle t \rangle$ is a subgroup of the additive group of $R$ and $|\langle t \rangle |=|t|=|R|$, it follows that $R$ is a cyclic ring.\end{proof}