File:Socle-of-module.svg

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Description
English: The socle of a module. This diagram represents it is the sum of minimal submodules and the intersection of large submodules. See Anderson-Fuller [Rings and Categories of Modules, second ed., (1992), Proposition 9.7].
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] {\( V \)};

% large submodules of V
\fill               (3, 6) [shade]  ellipse (120pt and 30pt);
\node (large1)   at (0, 6) [module] {};
\node (large2)   at (3, 6) [dammy]  {\( \cdots \)};
\node (large3)   at (6, 6) [module] {};
\node            at (9, 6)          {large submodules};

% socle of V
\node (socle)    at (3, 4) [module, color=red] {\( \operatorname{soc}V \)};

% minimal submodules of V
\fill               (3, 2) [shade]  ellipse (120pt and 30pt);
\node (minimal1) at (0, 2) [module] {};
\node (minimal2) at (3, 2) [dammy]  {\( \cdots \)};
\node (minimal3) at (6, 2) [module] {};
\node            at (9, 2)          {minimal submodules};

% zero
\node (zero)     at (3, 0) [module] {\( 0 \)};

% lines
\draw (whole)    -- (large1);
\draw (whole)    -- (large2);
\draw (whole)    -- (large3);

\draw (large1)   -- (socle);
\draw (large2)   -- (socle);
\draw (large3)   -- (socle);

\draw (socle)    -- (minimal1);
\draw (socle)    -- (minimal2);
\draw (socle)    -- (minimal3);

\draw (minimal1) -- (zero);
\draw (minimal2) -- (zero);
\draw (minimal3) -- (zero);
\end{tikzpicture}
\end{document}

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current09:04, 21 March 2016Thumbnail for version as of 09:04, 21 March 2016422 × 321 (27 KB)ARAKI Satoru (talk | contribs)User created page with UploadWizard

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