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href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="monoid_theory">Monoid theory</h4> <div class="hide"><div> <p><strong>monoid theory</strong> in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/infinity-monoid">infinity-monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object+in+an+%28infinity%2C1%29-category">monoid object in an (infinity,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a>, <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mon">Mon</a>, <a class="existingWikiWord" href="/nlab/show/CMon">CMon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+homomorphism">monoid homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+monoid">trivial monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/submonoid">submonoid</a>, <span class="newWikiWord">quotient monoid<a href="/nlab/new/quotient+monoid">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/divisor">divisor</a>, <span class="newWikiWord">multiple<a href="/nlab/new/multiple">?</a></span>, <span class="newWikiWord">quotient element<a href="/nlab/new/quotient+element">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a>, <a class="existingWikiWord" href="/nlab/show/unit">unit</a>, <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible element</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/principal+ideal+in+a+monoid">principal ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tensor+product+of+commutative+monoids">tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+monoid">GCD monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unique+factorization+monoid">unique factorization monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+monoid">Bézout monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+monoid">principal ideal monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/zero+divisor">zero divisor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+monoid">integral monoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a>, <a class="existingWikiWord" href="/nlab/show/free+commutative+monoid">free commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graphic+monoid">graphic monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+action">monoid action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+monoid">localization of a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+monoid">endomorphism monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/monoid+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <blockquote> <p>For other kinds of units see also <em><a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a></em> and <em><a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit of a monad</a></em>. Different (but related) is <em><a class="existingWikiWord" href="/nlab/show/physical+unit">physical unit</a></em>.</p> </blockquote> <hr /> <h1 id="units">Units</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#units_in_rings'>Units in rings</a></li> <li><a href='#units_in_monoids'>Units in monoids</a></li> <li><a href='#units_in_rngs_or_semigroups'>Units in rngs or semigroups</a></li> <li><a href='#units_in_nonassocative_rings_or_magmas'>Units in nonassocative rings or magmas</a></li> <li><a href='#units_in_modules'>Units in modules</a></li> <li><a href='#UnitsOfMeasurement'>Units of measurement</a></li> </ul> <li><a href='#identities_as_units'>Identities as units</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Considering a <a class="existingWikiWord" href="/nlab/show/ring">ring</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>, then by <em>the unit element</em> or <em>the multiplicative unit</em> one usually means the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">1 \in R</annotation></semantics></math> with respect to <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>. This is the sense of “unit” in terms such as <em><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></em>.</p> <p>But more generally <em>a unit element</em> in a unital (!) ring is any element that has an <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse element</a> under <a class="existingWikiWord" href="/nlab/show/multiplication">multiplication</a>.</p> <p>This concept generalizes beyond <a class="existingWikiWord" href="/nlab/show/rings">rings</a>, and this is what is discussed in the following.</p> <h2 id="definitions">Definitions</h2> <p>Exactly what this means depends on context. A very general definition is this:</p> <p>Given <a class="existingWikiWord" href="/nlab/show/sets">sets</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> and <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>, and a <a class="existingWikiWord" href="/nlab/show/function">function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">{\cdot}\colon R \times M \to M</annotation></semantics></math>, an <a class="existingWikiWord" href="/nlab/show/element">element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> of <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 a <strong>unit</strong> (relative to the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">{\cdot}</annotation></semantics></math>) if, given any 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> of <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>, there exists a unique element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">x = a \cdot u</annotation></semantics></math>.</p> <p>That is, every element of <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 a multiple (in a unique way) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>, where ‘multiple’ is defined in terms of the operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo></mrow><annotation encoding="application/x-tex">{\cdot}</annotation></semantics></math>.</p> <h3 id="units_in_rings">Units in rings</h3> <p>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 <a class="existingWikiWord" href="/nlab/show/ring">ring</a> (or <a class="existingWikiWord" href="/nlab/show/rig">rig</a>), then <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> comes equipped with a multiplication map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>×</mo><mi>R</mi><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">{\cdot}\colon R \times R \to R</annotation></semantics></math>. So <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> can play the role of both <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> and <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> above, although there are two ways to do this: on the left and on the right.</p> <p>We find that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <strong>left unit</strong> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/left+inverse">left inverse</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <strong>right unit</strong> if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+inverse">right inverse</a>. First, an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> with an inverse is a unit because, given any 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>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>u</mi></mrow><annotation encoding="application/x-tex"> x = (x u^{-1}) u </annotation></semantics></math></div> <p>(on the left) or</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>u</mi><mo stretchy="false">(</mo><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> x = u (u^{-1} x) </annotation></semantics></math></div> <p>(on the right). Conversely, a unit must have an inverse, since there must a solution to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>=</mo><mi>a</mi><mi>u</mi></mrow><annotation encoding="application/x-tex"> 1 = a u </annotation></semantics></math></div> <p>(on the left) or</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>=</mo><mi>u</mi><mi>a</mi></mrow><annotation encoding="application/x-tex"> 1 = u a </annotation></semantics></math></div> <p>(on the right).</p> <p>The collection of all units in a unital ring form a <a class="existingWikiWord" href="/nlab/show/group">group</a>, the <em><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></em>.</p> <p>In a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (or rig), a <strong>unit</strong> is an element 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> that has an <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse</a>, period. Of course, a commutative 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> is a <a class="existingWikiWord" href="/nlab/show/field">field</a> just when every non-<a class="existingWikiWord" href="/nlab/show/zero">zero</a> element is a unit.</p> <h3 id="units_in_monoids">Units in monoids</h3> <p>Notice that addition plays no role in the characterisation above of a unit in a ring. Accordingly, a unit in a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> may be defined in precisely the same way.</p> <p>A <a class="existingWikiWord" href="/nlab/show/group">group</a> is precisely a monoid in which every element is a unit.</p> <h3 id="units_in_rngs_or_semigroups">Units in rngs or semigroups</h3> <p>In a <a class="existingWikiWord" href="/nlab/show/rng">rng</a> (or, ignoring addition, in a <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a>), we cannot speak of inverses of elements. However, we can still talk about units; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <strong>left unit</strong> if, 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>, there is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>u</mi><mo>;</mo></mrow><annotation encoding="application/x-tex"> x = a u ;</annotation></semantics></math></div> <p>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is a <strong>right unit</strong> if, 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>, there is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>u</mi><mi>a</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> x = u a .</annotation></semantics></math></div> <h3 id="units_in_nonassocative_rings_or_magmas">Units in nonassocative rings or magmas</h3> <p>In a <a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative ring</a> (or, ignoring addition, in a <a class="existingWikiWord" href="/nlab/show/magma">magma</a>), even if we have an identity element, an invertible element might not be a unit. So we must use the same explicit definition as in a rng (or semigroup) above.</p> <p>A <a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a> is precisely a magma in which every element is a two-sided unit.</p> <h3 id="units_in_modules">Units in modules</h3> <p>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 <a class="existingWikiWord" href="/nlab/show/ring">ring</a> (or <a class="existingWikiWord" href="/nlab/show/rig">rig</a>) and <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> 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/module">module</a>, then a <strong>unit</strong> 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> is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">u \in M</annotation></semantics></math> such that every other <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">x \in M</annotation></semantics></math> can be written as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>a</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">x = a u</annotation></semantics></math> (or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>=</mo><mi>u</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">x = u a</annotation></semantics></math> for a right module) for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a \in R</annotation></semantics></math>. This is the same as a <a class="existingWikiWord" href="/nlab/show/generator">generator</a> of <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> 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>-module. There is no need to distinguish left and right units unless <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 a <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>. Note that a (left or right) unit 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> <em>qua</em> ring is the same as a unit 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> <em>qua</em> (left or 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>-module.</p> <h3 id="UnitsOfMeasurement">Units of measurement</h3> <p>In <a class="existingWikiWord" href="/nlab/show/physics">physics</a>, the quantities of a given dimension generally form 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">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/line">line</a>, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>-dimensional <a class="existingWikiWord" href="/nlab/show/real+vector+space">real vector space</a>. Since <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">\mathbb{R}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/field">field</a>, any non-<a class="existingWikiWord" href="/nlab/show/zero">zero</a> element is a unit, called in this context a <strong><a class="existingWikiWord" href="/nlab/show/unit+of+measurement">unit of measurement</a></strong>. This is actually a special case of a unit in a module, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>≔</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">R \coloneqq \mathbb{R}</annotation></semantics></math> and <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 the line in question.</p> <p>Often (but not always) these quantities form an <a class="existingWikiWord" href="/nlab/show/orientation">oriented</a> line, so that nonzero quantities are either positive or negative. Then we usually also require a unit of measurement to be positive. In fact, for some dimensions, there is no physical meaning to a negative quantity, in which case the quantities actually form a module over the rig <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}_{\ge 0}</annotation></semantics></math> and every nonzero element is “positive.”</p> <p>For example, the <a class="existingWikiWord" href="/nlab/show/kilogram">kilogram</a> is a unit of mass, because any mass may be expressed as a real multiple of the kilogram. Further, it is a positive unit; the mass of any physical object is a nonnegative quantity (so that mass quantities actually form an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℝ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbb{R}_{\ge 0}</annotation></semantics></math>-module) and may be expressed as a nonnegative real multiple of the kilogram.</p> <h2 id="identities_as_units">Identities as units</h2> <p>Often the term ‘unit’ (or ‘unity’) is used as a synonym for ‘<a class="existingWikiWord" href="/nlab/show/identity+element">identity element</a>’, especially when this identity element is denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>. For example, a ‘ring with unit’ (or ‘ring with unity’) is a ring with an identity (used by authors who say ‘ring’ for a rng). Of course, a rng with identity has a unit, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> itself is a unit; conversely, a commutative rng with a unit must have an identity.</p> <div class="query"> <p>I haven't managed to find either a proof or a counterexample to the converse (in the noncommutative case): that a rng with a unit must have an identity.</p> <p>Response: 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 rng with a unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>, then every element uniquely factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>. In particular, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> itself does. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><mi>a</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">u = a u</annotation></semantics></math>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> unique. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is an identity.</p> <p>Reply: Why is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> an identity then? This works if the rng is commutative: given any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mi>u</mi></mrow><annotation encoding="application/x-tex">b u</annotation></semantics></math>, and then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>v</mi><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>b</mi><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mo stretchy="false">(</mo><mi>a</mi><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi><mi>u</mi><mo>=</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">a v = a (b u) = b (a u) = b u = v</annotation></semantics></math>. But without commutativity (and associativity), this doesn't work.</p> <p>Response: I believe it also works in the non-commutative case, but with a more complicated proof.<br /> Suppose <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 not-necessarily-commutative rng with a unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math>;<br /> first, observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is neither a left nor a right zero divisor, as the equations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">0=x\cdot u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><mi>u</mi><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">0=u\cdot x</annotation></semantics></math> both have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> as a solution, and that must be unique.<br /> Now, for any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a,b\in R</annotation></semantics></math> denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>a</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">{}_a u^{-1}</annotation></semantics></math> the unique element s.t. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><msub><mrow></mrow> <mi>a</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">a= {}_a u^{-1} \cdot u</annotation></semantics></math>, and denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">u^{-1}_b</annotation></semantics></math> the unique element s.t. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>=</mo><mi>u</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">b= u\cdot u^{-1}_b</annotation></semantics></math>;<br /> we want to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">u^{-1}_u= {}_u u^{-1}</annotation></semantics></math> and that is the identity 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>.<br /> First, notice that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow></mrow> <mi>a</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>u</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow></mrow> <mi>a</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>u</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex"> a\cdot u^{-1}_u= {}_a u^{-1}\cdot u \cdot u^{-1}_u= {}_a u^{-1} \cdot u = a</annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>b</mi><mo>=</mo><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>u</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mi>u</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>b</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><mi>b</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> {}_u u^{-1} \cdot b= {}_u u^{-1} \cdot u\cdot u^{-1}_b=u\cdot u^{-1}_b=b,</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">a,b\in R</annotation></semantics></math>;<br /> therefore, we just need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">u^{-1}_u= {}_u u^{-1}</annotation></semantics></math>.<br /> To accomplish this, first notice that, on the one hand, one has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">u = {}_u u^{-1}\cdot u</annotation></semantics></math>;<br /> on the other hand, one has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>⋅</mo><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><mi>u</mi><mo>=</mo><mi>u</mi><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">u\cdot u^{-1}_u \cdot u=u\cdot u</annotation></semantics></math>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>⋅</mo><mo stretchy="false">(</mo><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><mi>u</mi><mo>−</mo><mi>u</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">u\cdot (u^{-1}_u\cdot u-u)=0</annotation></semantics></math>, which must imply <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">u= u^{-1}_u\cdot u</annotation></semantics></math> since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> is not a zero divisor.<br /> By uniqueness of the solution <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">u=x\cdot u</annotation></semantics></math>, we deduce that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>u</mi> <mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msub><mrow></mrow> <mi>u</mi></msub><msup><mi>u</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">u^{-1}_u= {}_u u^{-1}</annotation></semantics></math>.</p> <p>Addendum: Having uniqueness for the solutions is essential in order for the converse to hold.<br /> This is because, if a rng has an identity, the units are not zero divisors, therefore the equations of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><mi>u</mi></mrow><annotation encoding="application/x-tex">a=x\cdot u</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>=</mo><mi>u</mi><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">a=u\cdot x</annotation></semantics></math> all have a unique solution;<br /> therefore, by contrapositive, if some of these equations have multiple solutions, then the rng has no identity.</p> </div> <p>It is this meaning of ‘unit’ which gives rise to the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group+of+units">group of units</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+object">unit object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit of an adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unit+of+a+monad">unit of a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad+terminology">monad terminology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exponential+ring">exponential ring</a></p> </li> </ul> <h2 id="references">References</h2> <p>See also</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Unit_(ring_theory)">Unit (Ring theory)</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on September 25, 2024 at 10:12:24. See the <a href="/nlab/history/unit" 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/unit" accesskey="E" class="navlink" id="edit" rel="nofollow">Edit</a><a href="https://nforum.ncatlab.org/discussion/8491/#Item_5">Discuss</a><span class="backintime"><a href="/nlab/revision/unit/23" accesskey="B" class="navlinkbackintime" id="to_previous_revision" rel="nofollow">Previous revision</a></span><a href="/nlab/show/diff/unit" accesskey="C" class="navlink" id="see_changes" rel="nofollow">Changes from previous revision</a><a href="/nlab/history/unit" accesskey="S" class="navlink" id="history" rel="nofollow">History (23 revisions)</a> <a href="/nlab/show/unit/cite" style="color: black">Cite</a> <a href="/nlab/print/unit" accesskey="p" id="view_print" rel="nofollow">Print</a> <a href="/nlab/source/unit" id="view_source" rel="nofollow">Source</a> </div> </div>  </div>  </body> </html>