File:Radical-of-module.svg
Summary
Description |
English: The Jacobson radical of a module. This diagram represents it is the sum of small submodules and the intersection of maximal submodules. See Anderson-Fuller [Rings and Categories of Modules, second ed., (1992), Proposition 9.13]. |
Date | |
Source | Own work |
Author | ARAKI Satoru |
\documentclass[tikz]{standalone}
\usepackage{amsmath}
\begin{document}
\begin{tikzpicture}
[
module/.style = {circle, draw, minimum size=30pt},
dammy/.style = {minimum size=40pt},
shade/.style = {color=black!5},
]
% module V
\node (whole) at (3, 8) [module] {\( V \)};
% maximal submodules of V
\fill (3, 6) [shade] ellipse (120pt and 30pt);
\node (maximal1) at (0, 6) [module] {};
\node (maximal2) at (3, 6) [dammy] {\( \cdots \)};
\node (maximal3) at (6, 6) [module] {};
\node at (9, 6) {maximal submodules};
% radical of V
\node (radical) at (3, 4) [module, color=red] {\( \operatorname{rad}V \)};
% small submodules of V
\fill (3, 2) [shade] ellipse (120pt and 30pt);
\node (small1) at (0, 2) [module] {};
\node (small2) at (3, 2) [dammy] {\( \cdots \)};
\node (small3) at (6, 2) [module] {};
\node at (9, 2) {small submodules};
% zero
\node (zero) at (3, 0) [module] {\( 0 \)};
% lines
\draw (whole) -- (maximal1);
\draw (whole) -- (maximal2);
\draw (whole) -- (maximal3);
\draw (maximal1) -- (radical);
\draw (maximal2) -- (radical);
\draw (maximal3) -- (radical);
\draw (radical) -- (small1);
\draw (radical) -- (small2);
\draw (radical) -- (small3);
\draw (small1) -- (zero);
\draw (small2) -- (zero);
\draw (small3) -- (zero);
\end{tikzpicture}
\end{document}
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