PropertiesOfSemisimpleModules 2009-04-13 20:16:45 2009-04-13 20:16:45 Theorem completely reducible % 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 $R$ be a ring. Recall that $R$-module $M$ is called \textit{semisimple} iff $M$ is a direct sum of simple module. \textbf{Proposition.} The following are equivalent for $R$-module $M$:\begin{enumerate}\item $M$ is semisimple; \item $M$ is generated by its simple submodules; \item for every submodule $N\subseteq M$ there exists a submodule $N'\subseteq M$ such that $M=N\oplus N'$. \end{enumerate}