File:Radical-of-module.svg

From Wikimedia Commons, the free media repository
Jump to navigation Jump to search

Original file (SVG file, nominally 423 × 321 pixels, file size: 25 KB)

Captions

Captions

Add a one-line explanation of what this file represents

Summary

[edit]
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
 
W3C-validity not checked.
\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}

Licensing

[edit]
I, the copyright holder of this work, hereby publish it under the following license:
w:en:Creative Commons
attribution share alike
This file is licensed under the Creative Commons Attribution-Share Alike 4.0 International license.
You are free:
  • to share – to copy, distribute and transmit the work
  • to remix – to adapt the work
Under the following conditions:
  • attribution – You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use.
  • share alike – If you remix, transform, or build upon the material, you must distribute your contributions under the same or compatible license as the original.

File history

Click on a date/time to view the file as it appeared at that time.

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

There are no pages that use this file.

File usage on other wikis

The following other wikis use this file:

Metadata