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Reference.org
Double centralizer theorem
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<p>In the branch of <a href="/facts/Abstract_algebra/YNNE1CNz">abstract algebra</a> called <a href="/facts/Ring_theory/VW6svEFe">ring theory</a>, the double centralizer theorem can refer to any one of several similar results. These results concern the <a href="/facts/Centralizer_and_normalizer/AKhrEUGR">centralizer of a subring</a> <i>S</i> of a ring <i>R</i>, denoted C<i>R</i>(<i>S</i>) in this article. It is always the case that C<i>R</i>(C<i>R</i>(<i>S</i>)) contains <i>S</i>, and a double centralizer theorem gives conditions on <i>R</i> and <i>S</i> that guarantee that C<i>R</i>(C<i>R</i>(<i>S</i>)) is <i>equal</i> to <i>S</i>. </p>