ebab7d0da53c51438720d60c6e2e467c - /page/wikibook/entry.mu

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<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Zero element</span></span>

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</style><div role="note" class="hatnote navigation-not-searchable">For other uses of "Zero", see <a href="Zero_(disambiguation)" class="mw-disambig" title="Zero (disambiguation)">Zero (disambiguation)</a>.</div>

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<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, a <b>zero element</b> is one of several generalizations of <a href="0" title="0">the number zero</a> to other <a href="Algebraic_structure" title="Algebraic structure">algebraic structures</a>. These alternate meanings may or may not reduce to the same thing, depending on the context.

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<div class="mw-heading mw-heading2"><h2 id="Additive_identities">Additive identities</h2></div>

<p>An <i><a href="Additive_identity" title="Additive identity">additive identity</a></i> is the <a href="Identity_element" title="Identity element">identity element</a> in an <a href="Abelian_group" title="Abelian group">additive group</a> or <a href="Monoid" title="Monoid">monoid</a>. It corresponds to the element 0 such that for all x in the group, <span class="nowrap">0 + <i>x</i> = <i>x</i> + 0 = <i>x</i></span>. Some examples of additive identity include:

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<ul><li>The <b>zero vector</b> under <a href="Vector_addition" class="mw-redirect" title="Vector addition">vector addition</a>: the vector whose components are all 0; in a <a href="Normed_vector_space" title="Normed vector space">normed vector space</a> its norm (length) is also 0. Often denoted as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {0} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathbf {0} }</annotation>

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</math></span><img src="./62e8c650763635a93ddc69768c3c0c100afe985d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.337ex; height:2.176ex;" alt="{\displaystyle \mathbf {0} }" loading="lazy"></span> or <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {0}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {0}}}</annotation>

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</math></span><img src="./e76498919cf387316fc79d04120c59a8d430ef36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.162ex; height:2.843ex;" alt="{\displaystyle {\vec {0}}}" loading="lazy"></span>.<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></li>

<li>The <b>zero function</b> or <b>zero map</b> defined by <span class="nowrap"><i>z</i>(<i>x</i>) = 0</span>, under <a href="Pointwise" title="Pointwise">pointwise addition</a> <span class="nowrap">(<i>f</i> + <i>g</i>)(<i>x</i>) = <i>f</i>(<i>x</i>) + <i>g</i>(<i>x</i>)</span></li>

<li>The <i><a href="Empty_set" title="Empty set">empty set</a></i> under <a href="Union_(set_theory)" title="Union (set theory)">set union</a></li>

<li>An <i><a href="Empty_sum" title="Empty sum">empty sum</a></i> or <i>empty <a href="Coproduct" title="Coproduct">coproduct</a></i></li>

<li>An <i><a href="Initial_and_terminal_objects" title="Initial and terminal objects">initial object</a></i> in a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> (an empty coproduct, and so an identity under <a href="Coproduct" title="Coproduct">coproducts</a>)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Absorbing_elements">Absorbing elements</h2></div>

<p>An <i><a href="Absorbing_element" title="Absorbing element">absorbing element</a></i> in a multiplicative <a href="Semigroup" title="Semigroup">semigroup</a> or <a href="Semiring" title="Semiring">semiring</a> generalises the property <span class="nowrap">0 ⋅ <i>x</i> = 0</span>. Examples include:

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<ul><li>The <i><a href="Empty_set" title="Empty set">empty set</a></i>, which is an absorbing element under <a href="Cartesian_product" title="Cartesian product">Cartesian product</a> of sets, since <span class="nowrap">{ } × <i>S</i> = { }</span></li>

<li>The <b>zero function</b> or <b>zero map</b> defined by <span class="nowrap"><i>z</i>(<i>x</i>) = 0</span> under <a href="Pointwise" title="Pointwise">pointwise multiplication</a> <span class="nowrap">(<i>f</i> ⋅ <i>g</i>)(<i>x</i>) = <i>f</i>(<i>x</i>) ⋅ <i>g</i>(<i>x</i>)</span></li></ul>

<p>Many absorbing elements are also additive identities, including the empty set and the zero function. Another important example is the distinguished element 0 in a <i><a href="Field_(mathematics)" title="Field (mathematics)">field</a></i> or <i><a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a></i>, which is both the additive identity and the multiplicative absorbing element, and whose <a href="Principal_ideal" title="Principal ideal">principal ideal</a> is the smallest ideal.

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<div class="mw-heading mw-heading2"><h2 id="Zero_objects">Zero objects</h2></div>

<p>A <i><a href="Zero_object" class="mw-redirect" title="Zero object">zero object</a></i> in a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> is both an <a href="Initial_and_terminal_objects" title="Initial and terminal objects">initial and terminal object</a> (and so an identity under both <a href="Coproduct" title="Coproduct">coproducts</a> and <a href="Product_(category_theory)" title="Product (category theory)">products</a>). For example, the trivial structure (containing only the identity) is a zero object in categories where morphisms must map identities to identities. Specific examples include:

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<ul><li>The <i><a href="Trivial_group" title="Trivial group">trivial group</a></i>, containing only the identity (a zero object in the <a href="Category_of_groups" title="Category of groups">category of groups</a>)</li>

<li>The <b>zero module</b>, containing only the identity (a zero object in the category of <a href="Module_(mathematics)" title="Module (mathematics)">modules</a> over a ring)</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Zero_morphisms">Zero morphisms</h2></div>

<p>A <i><a href="Zero_morphism" title="Zero morphism">zero morphism</a></i> in a <a href="Category_(mathematics)" title="Category (mathematics)">category</a> is a generalised absorbing element under <a href="Function_composition" title="Function composition">function composition</a>: any morphism composed with a zero morphism gives a zero morphism. Specifically, if <span class="nowrap">0<sub><i>XY</i></sub> : <i>X</i> → <i>Y</i></span> is the zero morphism among morphisms from <i>X</i> to <i>Y</i>, and <span class="nowrap"><i>f</i> : <i>A</i> → <i>X</i></span> and <span class="nowrap"><i>g</i> : <i>Y</i> → <i>B</i></span> are arbitrary morphisms, then <span class="nowrap"><i>g</i> ∘ 0<sub><i>XY</i></sub> = 0<sub><i>XB</i></sub></span> and <span class="nowrap">0<sub><i>XY</i></sub> ∘ <i>f</i> = 0<sub><i>AY</i></sub></span>.

</p><p>If a category has a zero object <b>0</b>, then there are canonical morphisms <span class="nowrap"><i>X</i> → <b>0</b></span> and <span class="nowrap"><b>0</b> → <i>Y</i>,</span> and composing them gives a zero morphism <span class="nowrap">0<sub><i>XY</i></sub> : <i>X</i> → <i>Y</i></span>. In the <a href="Category_of_groups" title="Category of groups">category of groups</a>, for example, zero morphisms are morphisms which always return group identities, thus generalising the function <span class="nowrap"><i>z</i>(<i>x</i>) = 0.</span>

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<div class="mw-heading mw-heading2"><h2 id="Least_elements">Least elements</h2></div>

<p>A <i><a href="Least_element" class="mw-redirect" title="Least element">least element</a></i> in a <a href="Partially_ordered_set" title="Partially ordered set">partially ordered set</a> or <a href="Lattice_(order)" title="Lattice (order)">lattice</a> may sometimes be called a zero element, and written either as 0 or ⊥.

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<div class="mw-heading mw-heading2"><h2 id="Zero_module">Zero module</h2></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>zero module</b> is the <a href="Module_(mathematics)" title="Module (mathematics)">module</a> consisting of only the additive <a href="Identity_element" title="Identity element">identity</a> for the module's <a href="Addition" title="Addition">addition</a> function. In the <a href="Integer" title="Integer">integers</a>, this identity is <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a>, which gives the name <i>zero module</i>. That the zero module is in fact a module is simple to show; it is closed under addition and <a href="Multiplication" title="Multiplication">multiplication</a> trivially.

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<div class="mw-heading mw-heading2"><h2 id="Zero_ideal">Zero ideal</h2></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>zero ideal</b> in a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

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</math></span><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></span> is the <a href="Ideal_(ring_theory)" title="Ideal (ring theory)">ideal</a> <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{0\}}">

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</math></span><img src="./0ff0df9ef65c0572eb676580ce1c02b8ec40f694.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:3.487ex; height:2.843ex;" alt="{\displaystyle \{0\}}" loading="lazy"></span> consisting of only the additive identity (or <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a> element). The fact that this is an ideal follows directly from the definition.

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<div class="mw-heading mw-heading2"><h2 id="Zero_matrix">Zero matrix</h2></div>

<div role="note" class="hatnote navigation-not-searchable">Main article: <a href="Zero_matrix" title="Zero matrix">Zero matrix</a></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, particularly <a href="Linear_algebra" title="Linear algebra">linear algebra</a>, a <i><a href="Zero_matrix" title="Zero matrix">zero matrix</a></i> is a <a href="Matrix_(mathematics)" title="Matrix (mathematics)">matrix</a> with all its entries being <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a>. It is alternately denoted by the symbol <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

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<annotation encoding="application/x-tex">{\displaystyle O}</annotation>

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</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span>.<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> Some examples of zero matrices are

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<annotation encoding="application/x-tex">{\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\ }</annotation>

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</math></span><img src="./ec3480a57ed29cef2351da1ebb344475b1cccf13.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:47.028ex; height:6.176ex;" alt="{\displaystyle 0_{1,1}={\begin{bmatrix}0\end{bmatrix}},\ 0_{2,2}={\begin{bmatrix}0&0\\0&0\end{bmatrix}},\ 0_{2,3}={\begin{bmatrix}0&0&0\\0&0&0\end{bmatrix}},\ }" loading="lazy"></span></dd></dl>

<p>The set of <i>m</i> × <i>n</i> matrices with entries in a <a href="Ring_(mathematics)" title="Ring (mathematics)">ring</a> <i>K</i> forms a module <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{m,n}}">

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</math></span><img src="./72acfc2cdfe30839be0b076060dcc4d938596b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.092ex; height:2.843ex;" alt="{\displaystyle K_{m,n}}" loading="lazy"></span>. The zero matrix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{K_{m,n}}}">

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</math></span><img src="./fdb46270b31cdc1fe23cfdb7dc841839f347ef82.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:5.297ex; height:3.009ex;" alt="{\displaystyle 0_{K_{m,n}}}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{m,n}}">

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</math></span><img src="./72acfc2cdfe30839be0b076060dcc4d938596b53.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:5.092ex; height:2.843ex;" alt="{\displaystyle K_{m,n}}" loading="lazy"></span> is the matrix with all entries equal to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0_{K}}">

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<annotation encoding="application/x-tex">{\displaystyle 0_{K}}</annotation>

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</math></span><img src="./2eb8a92b2536158b176450bd0e3d43043539c0c8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.856ex; height:2.509ex;" alt="{\displaystyle 0_{K}}" loading="lazy"></span> is the additive identity in <i>K</i>.

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<annotation encoding="application/x-tex">{\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}}</annotation>

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</math></span><img src="./a420275fa4d87d6c1d06cd427a42fc59ce0fdaf0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.338ex; width:30.506ex; height:13.843ex;" alt="{\displaystyle 0_{K_{m,n}}={\begin{bmatrix}0_{K}&0_{K}&\cdots &0_{K}\\0_{K}&0_{K}&\cdots &0_{K}\\\vdots &\vdots &&\vdots \\0_{K}&0_{K}&\cdots &0_{K}\end{bmatrix}}}" loading="lazy"></span></dd></dl>

<p>The zero matrix is the additive identity in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K_{m,n}}">

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</math></span><img src="./ec0d39b6a1184b5ab9451a20d33c5dd6d66b23ea.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:9.675ex; height:2.843ex;" alt="{\displaystyle A\in K_{m,n}}" loading="lazy"></span>:

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<annotation encoding="application/x-tex">{\displaystyle 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A}</annotation>

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</math></span><img src="./b3b44d992e058550eadd091b4b06d3c657354313.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:27.702ex; height:3.009ex;" alt="{\displaystyle 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A}" loading="lazy"></span></dd></dl>

<p>There is exactly one zero matrix of any given size <i>m</i> × <i>n</i> (with entries from a given ring), so when the context is clear, one often refers to <i>the</i> zero matrix. In a <a href="Matrix_ring" title="Matrix ring">matrix ring</a>, the zero matrix serves the role of both an additive identity and an absorbing element. In general, the zero element of a ring is unique, and typically denoted as 0 without any subscript to indicate the parent ring. Hence the examples above represent zero matrices over any ring.

</p><p>The zero matrix also represents the <a href="Linear_transformation" class="mw-redirect" title="Linear transformation">linear transformation</a> which sends all vectors to the zero vector.

</p>

<div class="mw-heading mw-heading2"><h2 id="Zero_tensor">Zero tensor</h2></div>

<p>In <a href="Mathematics" title="Mathematics">mathematics</a>, the <b>zero tensor</b> is a <a href="Tensor" title="Tensor">tensor</a>, of any order, all of whose components are <a href="0_(number)" class="mw-redirect" title="0 (number)">zero</a>. The zero tensor of order 1 is sometimes known as the zero vector.

</p><p>Taking a <a href="Tensor_product" title="Tensor product">tensor product</a> of any tensor with any zero tensor results in another zero tensor. Among tensors of a given type, the zero tensor of that type serves as the additive identity among those tensors.

</p>

<ul><li><a href="Null_semigroup" title="Null semigroup">Null semigroup</a></li>

<li><a href="Zero_divisor" title="Zero divisor">Zero divisor</a></li>

<li><a href="Zero_object" class="mw-redirect" title="Zero object">Zero object</a></li>

<li><a href="Zero_of_a_function" title="Zero of a function">Zero of a function</a></li>

<li><a href="Zero" class="mw-redirect" title="Zero">Zero</a> — non-mathematical uses</li></ul>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

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</style><cite id="CITEREFNairSingh2018" class="citation book cs1">Nair, M. Thamban; Singh, Arindama (2018). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=WW1lDwAAQBAJ&pg=PA3"><i>Linear Algebra</i></a>. Springer. p. 3. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2018lial.book.....N">2018lial.book.....N</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2F978-981-13-0926-7">10.1007/978-981-13-0926-7</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-981-13-0925-0</bdi>.</cite></span>

</li>

<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text"><cite id="CITEREFLang1987" class="citation book cs1"><a href="Serge_Lang" title="Serge Lang">Lang, Serge</a> (1987). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=0DUXym7QWfYC&pg=PA25"><i>Linear Algebra</i></a>. <a href="Undergraduate_Texts_in_Mathematics" title="Undergraduate Texts in Mathematics">Undergraduate Texts in Mathematics</a>. Springer. p. 25. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780387964126</bdi>. <q>We have a zero matrix in which <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a_{ij}=0}">

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</math></span><img src="./a565e93211d1ad5c06a571ff8952ab3dfcff3638.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.968ex; height:2.843ex;" alt="{\displaystyle a_{ij}=0}" loading="lazy"></span> for all <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,j}">

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</math></span><img src="./f4cbf8bbc622154cda8208d6e339495fe16a1f9a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.794ex; height:2.509ex;" alt="{\displaystyle i,j}" loading="lazy"></span>. ... We shall write it <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle O}">

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</math></span><img src="./9d70e1d0d87e2ef1092ea1ffe2923d9933ff18fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.773ex; height:2.176ex;" alt="{\displaystyle O}" loading="lazy"></span>.</q></cite></span>

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