FreeModule3 2003-11-10 17:40:28 2004-04-28 23:47:13 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 Let $R$ be a ring. A {\it free module} over $R$ is a direct sum of copies of $R$. Similarly, as an abelian group is simply a module over $\Bbb{Z}$, a {\it free abelian group} is a direct sum of copies of $\Bbb{Z}$. This is equivalent to saying that the module has a {\it free basis}, i.e. a set of elements with the property that every element of the module can be uniquely expressed as an linear combination over $R$ of elements of the free basis. Every free module is also a projective module, as well as a flat module.