https://commalg.subwiki.org/w/index.php?action=history&feed=atom&title=Going-down_subringGoing-down subring - Revision history2026-04-29T02:43:00ZRevision history for this page on the wikiMediaWiki 1.41.2https://commalg.subwiki.org/w/index.php?title=Going-down_subring&diff=384&oldid=prevVipul: 1 revision2008-05-12T16:22:30Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://commalg.subwiki.org/w/index.php?title=Going-down_subring&diff=383&oldid=prevVipul at 09:55, 8 August 20072007-08-08T09:55:43Z<p></p>
<p><b>New page</b></p><div>{{curing-subring property}}<br />
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==Definition==<br />
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===Definition with symbols===<br />
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Suppose <math>S</math> is a unital subring of a [[commutative unital ring]] <math>R</math>. We say that <math>S</math> is a '''going-down subring''' if given [[prime idael]]s <math>Q \subseteq Q_1</math> of <math>S</math> and a prime ideal <math>P_1</math> of <math>R</math> lying over <math>Q_1</math> (viz <math>P_1 \cap S = Q_1</math>, there exists a prime <math>P</math> lying over <math>Q</math> (viz <math>P \cap S = Q</math> and contained in <math>P_1</math>.<br />
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==Facts==<br />
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If <math>S</math> is a [[normal domain]] and <math>R</math> is an [[integral extension]] of <math>S</math>, then <math>R</math> is a going-down subring.</div>Vipul