https://commalg.subwiki.org/w/index.php?action=history&feed=atom&title=Ideal_generated_by_an_irreducible_element 2026-04-22T15:30:40Z Revision history for this page on the wiki MediaWiki 1.41.2 https://commalg.subwiki.org/w/index.php?title=Ideal_generated_by_an_irreducible_element&diff=1932&oldid=prev 2009-02-04T17:38:35Z

<p>New page: {{curing-ideal property}} ==Definition== A proper nonzero <a href="/wiki/Ideal" title="Ideal">ideal</a> in a <a href="/wiki/Commutative_unital_ring" title="Commutative unital ring">commutative unital ring</a> is termed an &#039;&#039;&#039;ideal generated by an irreducible element if it satisfies the followi...</p> <p><b>New page</b></p><div>{{curing-ideal property}}<br /> <br /> ==Definition==<br /> <br /> A proper nonzero [[ideal]] in a [[commutative unital ring]] is termed an &#039;&#039;&#039;ideal generated by an irreducible element if it satisfies the following equivalent conditions:<br /> <br /> * It is the [[principal ideal]] generated by an [[fact about::irreducible element]].<br /> * It is a [[principal ideal]] and &#039;&#039;every&#039;&#039; element generating it is an irreducible element.<br /> <br /> ==Facts==<br /> <br /> * [[Irreducible element property is not determined by quotient ring]]: Whether a proper nonzero ideal &lt;math&gt;I&lt;/math&gt; in a ring &lt;math&gt;R&lt;/math&gt; is generated by an irreducible element cannot be determined by looking at the isomorphism type of the quotient ring &lt;math&gt;R/I&lt;/math&gt;.</div>

Vipul