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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Irrelevant ideal</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>irrelevant ideal</b> is the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> of a <a href="Graded_ring" title="Graded ring">graded ring</a> generated by the <a href="Homogeneous_element" class="mw-redirect" title="Homogeneous element">homogeneous elements</a> of degree greater than zero. It corresponds to the origin in the <a href="Affine_space" title="Affine space">affine space</a>, which cannot be mapped to a point in the <a href="Projective_space" title="Projective space">projective space</a>. More generally, a <a href="Homogeneous_ideal" class="mw-redirect" title="Homogeneous ideal">homogeneous ideal</a> of a graded ring is called an <b>irrelevant ideal</b> if its <a href="Radical_of_an_ideal" title="Radical of an ideal">radical</a> contains the irrelevant ideal.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup>
</p><p>The terminology arises from the connection with <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. If <i>R</i> = <i>k</i>[<i>x</i><sub>0</sub>, ..., <i>x<sub>n</sub></i>] (a <a href="Multivariate_polynomial_ring" class="mw-redirect" title="Multivariate polynomial ring">multivariate polynomial ring</a> in <i>n</i>+1 variables over an <a href="Algebraically_closed_field" title="Algebraically closed field">algebraically closed field</a> <i>k</i>) is graded with respect to <a href="Degree_of_a_polynomial" title="Degree of a polynomial">degree</a>, there is a <a href="Bijection" title="Bijection">bijective correspondence</a> between <a href="Projective_algebraic_set" class="mw-redirect" title="Projective algebraic set">projective algebraic sets</a> in <a href="Projective_space" title="Projective space">projective <i>n</i>-space over <i>k</i></a> and homogeneous, <a href="Radical_ideal" class="mw-redirect" title="Radical ideal">radical ideals</a> of <i>R</i> not equal to the irrelevant ideal; this is known as the <a href="Projective_Nullstellensatz" class="mw-redirect" title="Projective Nullstellensatz">projective Nullstellensatz</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> More generally, for an arbitrary graded ring <i>R</i>, the <a href="Proj_construction" title="Proj construction">Proj construction</a> disregards all irrelevant ideals of <i>R</i>.<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup>
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<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>
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<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><a href="#CITEREFZariskiSamuel1975">Zariski & Samuel 1975</a>, §VII.2, p. 154</span>
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<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, Exercise I.2.4</span>
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<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFHartshorne1977">Hartshorne 1977</a>, §II.2</span>
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<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
<ul><li>Sections 1.5 and 1.8 of <style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFEisenbud1995" class="citation cs2"><a href="David_Eisenbud" title="David Eisenbud">Eisenbud, David</a> (1995), <i>Commutative algebra with a view toward algebraic geometry</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 150, Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-94269-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1322960">1322960</a></cite></li>
<li><cite id="CITEREFHartshorne1977" class="citation cs2"><a href="Robin_Hartshorne" title="Robin Hartshorne">Hartshorne, Robin</a> (1977), <i><a href="Algebraic_Geometry_(book)" title="Algebraic Geometry (book)">Algebraic Geometry</a></i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 52, New York: Springer-Verlag, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90244-9</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0463157">0463157</a></cite></li>
<li><cite id="CITEREFZariskiSamuel1975" class="citation cs2"><a href="Oscar_Zariski" title="Oscar Zariski">Zariski, Oscar</a>; <a href="Pierre_Samuel" title="Pierre Samuel">Samuel, Pierre</a> (1975), <i>Commutative algebra volume II</i>, <a href="Graduate_Texts_in_Mathematics" title="Graduate Texts in Mathematics">Graduate Texts in Mathematics</a>, vol. 29 (Reprint of the 1960 ed.), Berlin, New York: <a href="Springer-Verlag" class="mw-redirect" title="Springer-Verlag">Springer-Verlag</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-387-90171-8</bdi>, <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=0389876">0389876</a></cite></li></ul>
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