https://commalg.subwiki.org/w/index.php?action=history&feed=atom&title=Primeness_is_contraction-closedPrimeness is contraction-closed - Revision history2026-04-23T10:09:44ZRevision history for this page on the wikiMediaWiki 1.41.2https://commalg.subwiki.org/w/index.php?title=Primeness_is_contraction-closed&diff=997&oldid=prevVipul: 1 revision2008-05-12T16:33:14Z<p>1 revision</p>
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</td></tr></table>Vipulhttps://commalg.subwiki.org/w/index.php?title=Primeness_is_contraction-closed&diff=996&oldid=prevVipul: New page: {{curing-ideal metaproperty satisfaction}} ==Statement== ===Property-theoretic statement=== The property of ideals in commutative unital rings of being a prime ideal satisfies t...2008-01-24T15:07:06Z<p>New page: {{curing-ideal metaproperty satisfaction}} ==Statement== ===Property-theoretic statement=== The <a href="/w/index.php?title=Property_of_ideals_in_commutative_unital_rings&action=edit&redlink=1" class="new" title="Property of ideals in commutative unital rings (page does not exist)">property of ideals in commutative unital rings</a> of being a <a href="/wiki/Prime_ideal" title="Prime ideal">prime ideal</a> satisfies t...</p>
<p><b>New page</b></p><div>{{curing-ideal metaproperty satisfaction}}<br />
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==Statement==<br />
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===Property-theoretic statement===<br />
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The [[property of ideals in commutative unital rings]] of being a [[prime ideal]] satisfies the [[metaproperty of ideals in commutative unital rings]] of being [[contraction-closed property of ideals in commutative unital rings|contraction-closed]].<br />
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===Verbal statement===<br />
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Given a [[homomorphism of commutative unital rings]], the [[contraction]] of a [[prime ideal]] in the ring on the right, is a prime ideal in the ring on the left.<br />
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===Symbolic statement===<br />
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Suppose <math>f:R \to S</math> is a [[homomorphism of commutative unital rings]]. Then for any [[prime ideal]] <math>I</math> of <math>S</math>, <math>I^c = f^{-1}(I)</math> (called the [[contraction]] of <math>I</math>) is a prime ideal of <math>R</math>.<br />
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==Importance==<br />
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This fact allows us to view the [[spectrum]] of a commutative unital ring as a contravariant functor, because it allows us to use a [[homomorphism of commutative unital rings]] <math>f:R \to S</math> to define a backward map <math>Spec(f): Spec(S) \to Spec(R)</math>, by contraction.<br />
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==Proof==<br />
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{{fillin}}</div>Vipul