https://commalg.subwiki.org/w/index.php?action=history&feed=atom&title=Unique_factorization_and_Dedekind_iff_principal_ideal 2026-08-11T06:27:12Z Revision history for this page on the wiki MediaWiki 1.41.2 https://commalg.subwiki.org/w/index.php?title=Unique_factorization_and_Dedekind_iff_principal_ideal&diff=2016&oldid=prev 2009-02-07T00:59:44Z

<p>New page: {{definition equivalence|principal ideal domain}} ==Statement== The following are equivalent for an <a href="/wiki/Integral_domain" title="Integral domain">integral domain</a>: * It is a <a href="/w/index.php?title=Fact_about::unique_factorization_domain&amp;action=edit&amp;redlink=1" class="new" title="Fact about::unique factorization domain (page does not exist)">fact about::unique factorization domain</a> as well as...</p> <p><b>New page</b></p><div>{{definition equivalence|principal ideal domain}}<br /> <br /> ==Statement==<br /> <br /> The following are equivalent for an [[integral domain]]:<br /> <br /> * It is a [[fact about::unique factorization domain]] as well as a [[fact about::Dedekind domain]].<br /> * It is a [[fact about::principal ideal domain]].<br /> <br /> ==Facts used==<br /> <br /> # [[uses::PID implies UFD]]<br /> # [[uses::PID implies Dedekind]]<br /> # [[uses::Dedekind implies one-dimensional]]<br /> # [[uses::Unique factorization and one-dimensional iff principal ideal]]<br /> <br /> ==Proof==<br /> <br /> ===Principal ideal implies UFD and Dedekind===<br /> <br /> This is facts (1) and (2).<br /> <br /> ===UFD and Dedekind implies principal ideal domain===<br /> <br /> This is facts (3) and (4).</div>

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