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Codimension of an ideal - Revision history
2026-11-05T04:11:48Z
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Vipul: 2 revisions
2008-05-12T16:19:12Z
<p>2 revisions</p>
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<td colspan="1" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 16:19, 12 May 2008</td>
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Vipul
https://commalg.subwiki.org/w/index.php?title=Codimension_of_an_ideal&diff=131&oldid=prev
Vipul at 00:46, 8 January 2008
2008-01-08T00:46:58Z
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 00:46, 8 January 2008</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>Let <math>R</math> be a [[commutative unital ring]] and <math>I</math> an [[ideal]] in <math>R</math>. The '''codimension''' of <math>I</math> is defined as follows:</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>Let <math>R</math> be a [[commutative unital ring]] and <math>I</math> an [[ideal]] in <math>R</math>. The '''codimension<ins style="font-weight: bold; text-decoration: none;">''' or '''height</ins>''' of <math>I</math> is defined as follows:</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>I</math> is a [[prime ideal]], it is defined as the [[Krull dimension]] of the [[localization at a prime ideal|localization]] <math>R_I</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* If <math>I</math> is a [[prime ideal]], it is defined as the [[Krull dimension]] of the [[localization at a prime ideal|localization]] <math>R_I</math></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Otherwise, it is defined to be the minimum of the Krull dimensions of [[prime ideal]]s containing <math>I</math></div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>* Otherwise, it is defined to be the minimum of the Krull dimensions of [[prime ideal]]s containing <math>I</math></div></td></tr>
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Vipul
https://commalg.subwiki.org/w/index.php?title=Codimension_of_an_ideal&diff=130&oldid=prev
Vipul at 00:23, 8 January 2008
2008-01-08T00:23:11Z
<p></p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
Let <math>R</math> be a [[commutative unital ring]] and <math>I</math> an [[ideal]] in <math>R</math>. The '''codimension''' of <math>I</math> is defined as follows:<br />
<br />
* If <math>I</math> is a [[prime ideal]], it is defined as the [[Krull dimension]] of the [[localization at a prime ideal|localization]] <math>R_I</math><br />
* Otherwise, it is defined to be the minimum of the Krull dimensions of [[prime ideal]]s containing <math>I</math></div>
Vipul