File:Radical-of-module.svg

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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
 
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\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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I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
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Date/TimeThumbnailDimensionsUserComment
current05:35, 21 March 2016Thumbnail for version as of 05:35, 21 March 2016423 × 321 (25 KB)ARAKI Satoru (talk | contribs)User created page with UploadWizard

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