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this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#in_rings_and_other_rigs'>In rings (and other rigs)</a></li> <li><a href='#in_lattices_and_other_prosets'>In lattices (and other prosets)</a></li> <li><a href='#in_both_at_once'>In both at once</a></li> <li><a href='#in_monoids'>In monoids</a></li> <li><a href='#in_categories'>In categories</a></li> <li><a href='#in_additive_categories'>In additive categories</a></li> </ul> <li><a href='#kinds_of_ideals'>Kinds of ideals</a></li> <li><a href='#related_concepts'>Related concepts</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>Ideals show up both in <a class="existingWikiWord" href="/nlab/show/ring">ring</a> theory and in <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> theory. We recall both of these below and look at some slight generalizations.</p> <h3 id="in_rings_and_other_rigs">In rings (and other rigs)</h3> <p>A <strong>left ideal</strong> in a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> (or even <a class="existingWikiWord" href="/nlab/show/rig">rig</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> of (the underlying set of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> such that:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0 \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x + y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x, y \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math>, regardless of whether <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math>.</li> </ul> <p>A <strong>right ideal</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> such that:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0 \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x + y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x, y \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math>.</li> </ul> <p>A <strong>two-sided ideal</strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> that is both a left and right ideal; that is:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">0 \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>+</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x + y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math>.</li> </ul> <p>This generalises to:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mo>+</mo><mi>⋯</mi><mo>+</mo><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x_1 + \cdots + x_n \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x_k \in I</annotation></semantics></math> for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mn>1</mn></msub><mi>⋯</mi><msub><mi>x</mi> <mi>n</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x_1 \cdots x_n \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>k</mi></msub><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x_k \in I</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</li> </ul> <p>Notice that all three kinds of ideal are equivalent for a commutative ring.</p> <div class="num_remark"> <h6 id="remarks">Remarks</h6> <ul> <li>A left ideal in a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> may be equivalently defined as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-submodule of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, viewing the latter as a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>.</li> <li>A right ideal in a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> may be equivalently defined as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-submodule of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, viewing the latter as a right <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>.</li> <li>A two-sided ideal in a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> may be equivalently defined as a sub-bimodule of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, viewing the latter as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>.</li> <li>The collection of all two-sided ideals in a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> form a <a class="existingWikiWord" href="/nlab/show/concrete+category">concrete category</a>, see <a class="existingWikiWord" href="/nlab/show/category+of+two-sided+ideals+in+a+ring">category of two-sided ideals in a ring</a></li> <li>The preceding remarks apply to rigs as well.</li> <li>Considering the category of rings as a Barr-exact category, there is a natural bijection between <a class="existingWikiWord" href="/nlab/show/congruence">congruence relations</a> on a ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> (internal to the category of rings) and two-sided ideals of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>; this associates to each ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> the relation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∼</mo> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\sim_I</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><msub><mo>∼</mo> <mi>I</mi></msub><mi>y</mi></mrow><annotation encoding="application/x-tex">x \sim_I y</annotation></semantics></math> means <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>−</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x - y \in I</annotation></semantics></math>. This observation does not apply to the category of rigs.</li> </ul> </div> <p>This definition also makes sense for <a class="existingWikiWord" href="/nlab/show/nonassociative+rings">nonassociative rings</a> (or rigs), with the left/right/two-sided coincidence in the commutative case.</p> <p>In the case of a <a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a> (or rig), if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is being thought of as a <a class="existingWikiWord" href="/nlab/show/module">module</a> over some other ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, then it won't immediately follow that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/submodule">submodule</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>, and so one usually includes that requirement as well:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">a x \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">a \in K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math></li> </ul> <p>in the case of left-modules, and similarly in the case of right modules or <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a>. (Technically, this should be distinguished in the terminology, say by calling <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> an <strong>ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math></strong> or the like.) In particular, a (real or complex) <a class="existingWikiWord" href="/nlab/show/Lie+ideal">Lie ideal</a> of a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is an ideal of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> over the real or complex field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <h3 id="in_lattices_and_other_prosets">In lattices (and other prosets)</h3> <p>An <strong>ideal</strong> in a <a class="existingWikiWord" href="/nlab/show/lattice">lattice</a> (or even <a class="existingWikiWord" href="/nlab/show/preorder">proset</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subset">subset</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> of (the underlying set of) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> such that:</p> <ul> <li>There is an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> (so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/inhabited+set">inhabited</a>);</li> <li>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x, y \in I</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>≤</mo><mi>z</mi></mrow><annotation encoding="application/x-tex">x, y \leq z</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">z \in I</annotation></semantics></math>;</li> <li>if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y \leq x</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math> too.</li> </ul> <p>We can make this look more algebraic if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a (bounded) join-<a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a>:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\bot \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∨</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \vee y \in I</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x, y \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∨</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \vee y \in I</annotation></semantics></math>.</li> </ul> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is indeed a lattice, then we can make this look just like the ring version:</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊥</mo><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\bot \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∨</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \vee y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x, y \in I</annotation></semantics></math>;</li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∧</mo><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \wedge y \in I</annotation></semantics></math> whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math>.</li> </ul> <p>The concept of ideal is dual to that of <a class="existingWikiWord" href="/nlab/show/filter">filter</a>. A subset of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> that satisfies the first two of the three axioms for an ideal in a proset is precisely a <a class="existingWikiWord" href="/nlab/show/direction">directed subset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math>; notice that this is weaker than being a sub-join-semilattice even if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a lattice.</p> <h3 id="in_both_at_once">In both at once</h3> <p>There are some common situations where these two kinds of ideal might seem to clash but fortunately do not:</p> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/distributive+lattice">distributive lattice</a> is both a lattice and a commutative rig; the two concepts of ideal are the same, as can be seen by comparing the definition for rigs to the last definition for lattices.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a> is a both a distributive lattice and a <a class="existingWikiWord" href="/nlab/show/Boolean+ring">Boolean ring</a>; again, the two concepts of ideal are the same (partly because the multiplication operators are the same, although there is still some checking to do regarding closure under addition).</p> </li> </ul> <p>On the other hand, every poset is a poset in an <a class="existingWikiWord" href="/nlab/show/opposite+poset">opposite</a> way, and this does <em>not</em> give the same concept of ideal; an ideal in one is a <a class="existingWikiWord" href="/nlab/show/filter">filter</a> in the opposite one. We are lucky that the convention for interpreting a Boolean ring as a lattice goes in the correct direction, or the two notions of ideal in a Boolean algebra would not match; or perhaps it is not a matter of luck, but the convention for which way to define ideals in a lattice was chosen precisely to match the conventions for Boolean algebras!</p> <h3 id="in_monoids">In monoids</h3> <p>There is a notion of ideal in a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (or even <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>), or more generally in a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> in any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, which generalises the notion of ideal in a ri(n)g or in a (semi)lattice. That is, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, then a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>; if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Ab+Mon">Ab Mon</a>, then a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/rig">rig</a>; and a <a class="existingWikiWord" href="/nlab/show/semilattice">semilattice</a> is a commutative idempotent monoid in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. A left (right) ideal in a monoid (or semigroup) is a subset closed under multiplications with arbitrary elements in the semigroup from the left (right). See <a class="existingWikiWord" href="/nlab/show/ideal+in+a+monoid">ideal in a monoid</a>.</p> <p>This generalizes all of the above notions of ideal <em>except</em> for ideals in prosets that are not (possibly unbounded) join-semilattices.</p> <h3 id="in_categories">In categories</h3> <p>More generally still, passing from monoids to their many-object version and from prosets to their many-morphism version, a right ideal or <em><a class="existingWikiWord" href="/nlab/show/sieve">sieve</a></em> in a <a class="existingWikiWord" href="/nlab/show/category">category</a> is a subcategory closed under precomposition with morphisms from the entire category, and a cosieve or left ideal is the dual notion closed under postcompositions with morphisms in the entire category. See <a class="existingWikiWord" href="/nlab/show/sieve">sieve</a>.</p> <h3 id="in_additive_categories">In additive categories</h3> <p>In ringoids (small additive categories) and additive categories in general, a left/right/2-sided ideal is just the ideal (sieve/cosieve) in the sense of categories which is also closed under group operations in hom groups.</p> <h2 id="kinds_of_ideals">Kinds of ideals</h2> <p>Ideals form <a class="existingWikiWord" href="/nlab/show/complete+lattices">complete lattices</a> where arbitrary meets are given by set-theoretic intersection. In other words, ideals form a <a class="existingWikiWord" href="/nlab/show/Moore+collection">Moore collection</a> of subsets of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a rig, or of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is a lattice. This implies we have an ideal <strong>generated</strong> by any <a class="existingWikiWord" href="/nlab/show/subset">subset</a>: the intersection of all ideals containing the subset. A subset <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> that generates a given ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> may be called a <strong><a class="existingWikiWord" href="/nlab/show/subbase">subbase</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>; then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/base">base</a></strong> if every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a multiple (in a rig) or a predecessor (in an order) of some element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>. (In particular, every <a class="existingWikiWord" href="/nlab/show/singleton+subset">singleton subset</a> is a base of its generated ideal.) See also <a class="existingWikiWord" href="/nlab/show/filter+base">filter base</a> and dualize for more about bases and subbases of ideals in lattices and other posets.</p> <p>Certain kinds of ideals are often characterized by the roles they play in ideal lattices, or in terms of the Moore <a class="existingWikiWord" href="/nlab/show/closure+operator">closure operator</a>. Some examples follow.</p> <p>The top element of an ideal lattice is called the <em>improper ideal</em>. That is to say, an ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is the improper ideal if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math> for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> (which follows if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">1 \in I</annotation></semantics></math> for the case of rigs, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\top \in I</annotation></semantics></math> for the case of bounded lattices). An ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <em>proper</em> if it is not the improper ideal: if there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∉</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \notin I</annotation></semantics></math>. So in a <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is proper iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∉</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">1 \notin I</annotation></semantics></math>; in a (bounded) lattice, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is proper iff <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>∉</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">\top \notin I</annotation></semantics></math>.</p> <p>An ideal is a <strong><a class="existingWikiWord" href="/nlab/show/maximal+ideal">maximal ideal</a></strong> if it is maximal among <em>proper</em> ideals. A maximal ideal in a rig (including in a distributive lattice, but not in every lattice) is necessarily <a class="existingWikiWord" href="/nlab/show/prime+ideal">prime</a>; conversely, a prime ideal in a <a class="existingWikiWord" href="/nlab/show/Boolean+algebra">Boolean algebra</a> is necessarily maximal.</p> <p>An ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <strong><a class="existingWikiWord" href="/nlab/show/principal+ideal">principal ideal</a></strong> if it is generated by a singleton. This means there exists an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x \in I</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math> is a multiple of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> (in a rig) or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>≤</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">y \leq x</annotation></semantics></math> (in an ordered set) whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">y \in I</annotation></semantics></math>; we say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <strong>generated</strong> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math>. Thus every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> generates a unique principal ideal, the set of all left/right/two-sided multiples of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>x</mi></mrow><annotation encoding="application/x-tex">a x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">x b</annotation></semantics></math>, or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>x</mi><mi>b</mi></mrow><annotation encoding="application/x-tex">a x b</annotation></semantics></math> if we are talking about left/right/two-sided ideals in a rig) or the <a class="existingWikiWord" href="/nlab/show/downset">downset</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> (in an an order). Clearly, every ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a join over all the principal ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">P_x</annotation></semantics></math> generated by the elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>.</p> <p>As discussed at <a class="existingWikiWord" href="/nlab/show/ideals+in+a+monoid">ideals in a monoid</a>, there is for two-sided ideals an operation of ideal multiplication, making the ideal lattice a <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a> (cf. <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a>). Namely, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I, J</annotation></semantics></math> are ideals, then their product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi></mrow><annotation encoding="application/x-tex">I J</annotation></semantics></math> is the ideal generated by all products <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mi>y</mi></mrow><annotation encoding="application/x-tex">x y</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">x \in I, y \in J</annotation></semantics></math> in the case of rigs. Similarly, in the case of lattices, we could define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi></mrow><annotation encoding="application/x-tex">I J</annotation></semantics></math> to be the ideal generated by all meets <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∧</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x \wedge y</annotation></semantics></math> – but in this case the result is the same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∩</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I \cap J</annotation></semantics></math>. In any case, we say that a proper ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> is <strong><a class="existingWikiWord" href="/nlab/show/prime+ideal">prime</a></strong> if for any ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I, J</annotation></semantics></math>, the condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi><mo>⊆</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">I J \subseteq P</annotation></semantics></math> implies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>⊆</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">I \subseteq P</annotation></semantics></math> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>⊆</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">J \subseteq P</annotation></semantics></math>. In <a class="existingWikiWord" href="/nlab/show/order+theory">order theory</a>, the term ‘prime ideal’ is usually used for a <em><a class="existingWikiWord" href="/nlab/show/strongly+irreducible+ideal">strongly irreducible ideal</a></em>, since the two are equivalent for prosets.</p> <p>An (left, right, or two-sided) ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is <strong><a class="existingWikiWord" href="/nlab/show/irreducible+ideal">irreducible</a></strong> if it is proper and, whenever it is written as the <a class="existingWikiWord" href="/nlab/show/intersection">intersection</a> of two ideals (of the same kind) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mi>J</mi><mo>∩</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">I = J \cap K</annotation></semantics></math>, then at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> equals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. If we replace equality here by containment, then we get something more analogous to the definition of prime ideal, called a <strong><a class="existingWikiWord" href="/nlab/show/strongly+irreducible+ideal">strongly irreducible ideal</a></strong>. (Indeed, for <a class="existingWikiWord" href="/nlab/show/prosets">prosets</a>, prime ideals and strongly irreducible ideals are the same.) In general, prime ideals are strongly irreducible, and strongly irreducible ideals are irreducible, but the converses fail. An ideal is <strong><a class="existingWikiWord" href="/nlab/show/completely+irreducible+ideal">completely irreducible</a></strong> if whenever <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is an intersection of any (possibly infinite) collection of ideals, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is equal to at least one of them. (This is <em>not</em> analogous to <a class="existingWikiWord" href="/nlab/show/completely+prime+ideals">completely prime ideals</a>.)</p> <p>An ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> is a <strong>nil ideal</strong> if for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x\in I</annotation></semantics></math>, there is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n\in \mathbb{N}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>n</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^n=0</annotation></semantics></math>. That is, every element of the ideal is nilpotent. If on the other hand, there is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n\in \mathbb{N}</annotation></semantics></math>, such that for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>I</mi></mrow><annotation encoding="application/x-tex">x\in I</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mi>n</mi></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^n=0</annotation></semantics></math>, the ideal is called <strong>nilpotent</strong>.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>A maximal ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is prime.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Because the ideal lattice is a <a class="existingWikiWord" href="/nlab/show/quantale">quantale</a>, multiplication of ideals distributes over ideal joins. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi><mo>⊆</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">I J \subseteq M</annotation></semantics></math> for two ideals <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>,</mo><mi>J</mi></mrow><annotation encoding="application/x-tex">I, J</annotation></semantics></math>. If neither is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>∨</mo><mi>M</mi><mo>=</mo><mo>⊤</mo><mo>=</mo><mi>J</mi><mo>∨</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">I \vee M = \top = J \vee M</annotation></semantics></math> (the improper ideal) since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is maximal. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>=</mo><mo>⊤</mo><mo>⋅</mo><mo>⊤</mo><mo>=</mo><mo stretchy="false">(</mo><mi>I</mi><mo>∨</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>J</mi><mo>∨</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mi>I</mi><mi>J</mi><mo>∨</mo><mi>M</mi><mi>J</mi><mo>∨</mo><mi>I</mi><mi>M</mi><mo>∨</mo><mi>M</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">\top = \top \cdot \top = (I \vee M) \cdot (J \vee M) = I J \vee M J \vee I M \vee M M</annotation></semantics></math></div> <p>where all four summands are contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mi>J</mi><mo>⊆</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">I J \subseteq M</annotation></semantics></math> by supposition, and the other containments hold since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is an ideal). Thus their join is contained in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, so we have proved <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊤</mo><mo>⊆</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\top \subseteq M</annotation></semantics></math>, contradiction.</p> </div> <p>That every ideal is contained in a prime ideal is a <a class="existingWikiWord" href="/nlab/show/prime+ideal+theorem">prime ideal theorem</a>; that every ideal is contained in a maximal ideal is a <a class="existingWikiWord" href="/nlab/show/maximal+ideal+theorem">maximal ideal theorem</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+ideal">differential ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/germ-determined+ideal">germ-determined ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+in+a+monoid">ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/filter+of+a+ring">filter of a ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+ideal">modular ideal</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+predicate">ideal predicate</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/anti-ideal">anti-ideal</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on October 1, 2024 at 19:40:16. See the <a href="/nlab/history/ideal" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="/nlab/edit/ideal" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8890/#Item_6">Discuss</a><span class="backintime"><a href="/nlab/revision/ideal/40" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/ideal" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/ideal" accesskey="S" class="navlink" id="history" rel="nofollow">History (40 revisions)</a> <a href="/nlab/show/ideal/cite" style="color: black">Cite</a> <a href="/nlab/print/ideal" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/ideal" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>