Principle Ideal Domains - Revision history

http://wiki.algebrist.ddns.net/index.php?action=history&feed=atom&title=Principle_Ideal_Domains Principle Ideal Domains - Revision history 2026-05-01T13:42:25Z Revision history for this page on the wiki MediaWiki 1.33.4 http://wiki.algebrist.ddns.net/index.php?title=Principle_Ideal_Domains&diff=79&oldid=prev Kfagerstrom at 01:12, 8 March 2023 2023-03-08T01:12:56Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 01:12, 8 March 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l14" >Line 14:</td> <td colspan="2" class="diff-lineno">Line 14:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>PID implies [[Noetherian Ring| Noetherian]].</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>PID implies [[Noetherian Ring| Noetherian]].</div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins style="font-weight: bold; text-decoration: none;">[[Category: Ring Theory]]</ins></div></td></tr>  </table>

Kfagerstrom http://wiki.algebrist.ddns.net/index.php?title=Principle_Ideal_Domains&diff=58&oldid=prev Kfagerstrom at 20:35, 17 January 2023 2023-01-17T20:35:11Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:35, 17 January 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l13" >Line 13:</td> <td colspan="2" class="diff-lineno">Line 13:</td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>In a PID, irreducible elements are prime. </div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>In a PID, irreducible elements are prime. </div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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;"></td></tr> <tr><td class='diff-marker'>−</td><td style="color: #222; 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>PID implies [[Noetherian|<del class="diffchange diffchange-inline">Notherian Rings</del>]].</div></td><td class='diff-marker'>+</td><td style="color: #222; 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>PID implies [[Noetherian <ins class="diffchange diffchange-inline">Ring</ins>| <ins class="diffchange diffchange-inline">Noetherian</ins>]].</div></td></tr>  </table>

Kfagerstrom http://wiki.algebrist.ddns.net/index.php?title=Principle_Ideal_Domains&diff=57&oldid=prev Kfagerstrom at 20:34, 17 January 2023 2023-01-17T20:34:35Z

<p></p> <table class="diff diff-contentalign-left" data-mw="interface"> <col class="diff-marker" /> <col class="diff-content" /> <col class="diff-marker" /> <col class="diff-content" /> <tr class="diff-title" lang="en"> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">← Older revision</td> <td colspan="2" style="background-color: #fff; color: #222; text-align: center;">Revision as of 20:34, 17 January 2023</td> </tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td> <td colspan="2" class="diff-lineno">Line 1:</td></tr> <tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color: #222; 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><ins style="font-weight: bold; text-decoration: none;">:''see also'' [[817 - Algebra#Rings| Rings]] </ins></div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>A ''Principal Ideal Domain'' (PID) is a</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>A ''Principal Ideal Domain'' (PID) is a</div></td></tr> <tr><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have <math>I = (a) </math> for some <math>a \in R</math>.</div></td><td class='diff-marker'> </td><td style="background-color: #f8f9fa; color: #222; 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>domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have <math>I = (a) </math> for some <math>a \in R</math>.</div></td></tr>  </table>

Kfagerstrom http://wiki.algebrist.ddns.net/index.php?title=Principle_Ideal_Domains&diff=56&oldid=prev Kfagerstrom: Created page with "A ''Principal Ideal Domain'' (PID) is a domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have <math>I = (a) </ma..." 2023-01-17T20:25:38Z

<p>Created page with "A ''Principal Ideal Domain'' (PID) is a domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have <math>I = (a) </ma..."</p> <p><b>New page</b></p><div>A ''Principal Ideal Domain'' (PID) is a<br /> domain, <math>R</math> with the property that every ideal is principal, i.e., for each ideal <math>I</math>, we have <math>I = (a) </math> for some <math>a \in R</math>.<br /> <br /> All PIDs are EDs, but the opposite is not true. For example the ring <math>\mathbb{Z}[\frac{1+\sqrt{-19}}{2}]=\{a+b\frac{1+\sqrt{-19}}{2} | a,b \in \mathbb{Z}\} </math><br /> <br /> If <math>R</math> is a PID with <math>a,b\in \R</math>, then:<br /> *<math>(a,b)=(g)</math> for some <math>g\in R </math> and any such <math>g</math> is a gcd of <math>a</math> and <math>b</math>.<br /> * The gcd of <math>a</math> and <math>b</math> is unique up to multiplication by a unit<br /> <br /> If R is not only a PID but a Euclidean domain with norm function <math>N</math>, then the Euclidean algorithm can be used to compute a gcd of any two nonzero <math>a, b \in R</math>.<br /> <br /> In a PID, irreducible elements are prime. <br /> <br /> PID implies [[Noetherian|Notherian Rings]].</div>

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