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About: Unit (ring theory)
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href="/sparql/" title="Switch to /sparql endpoint"><i class="bi-box-arrow-up-right"></i> Sparql Endpoint </a> </li> </ul> </div> </div> </nav> <div style="margin-bottom: 60px"></div> <!-- /navbar --> <!-- page-header --> <section> <div class="container-xl"> <div class="row"> <div class="col"> <h1 id="title" class="display-6"><b>About:</b> <a href="http://dbpedia.org/resource/Unit_(ring_theory)">Unit (ring theory)</a> </h1> </div> </div> <div class="row"> <div class="col"> <div class="text-muted"> <span class="text-nowrap">An Entity of Type: <a href="http://dbpedia.org/ontology/Person">person</a>, </span> <span class="text-nowrap">from Named Graph: <a href="http://dbpedia.org">http://dbpedia.org</a>, </span> <span class="text-nowrap">within Data Space: <a href="http://dbpedia.org">dbpedia.org</a></span> </div> </div> </div> <div class="row pt-2"> <div class="col-xs-9 col-sm-10"> <p class="lead">In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).</p> </div> </div> </div> </section> <!-- page-header --> <!-- property-table --> <section> <div class="container-xl"> <div class="row"> <div class="table-responsive"> <table class="table table-hover table-sm table-light"> <thead> <tr> <th class="col-xs-3 ">Property</th> <th class="col-xs-9 px-3">Value</th> </tr> </thead> <tbody> <tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/abstract"><small>dbo:</small>abstract</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ca" >En matemàtiques, un element invertible d'un conjunt amb una és aquell del qual es pot obtenir un element invers per aquesta llei. Si la llei és associativa, els elements invertibles del conjunt tenen cadascun un únic invers. En teoria d'anells, els elements invertibles per la segona llei de composició interna (o producte) d'un anell s'acostumen a anomenar unitats. Si tenim un anell amb element neutre (A, +, ⋅, 0, 1), es denota A* el conjunt de les unitats, que és un grup amb la multiplicació i s'anomena grup de les unitats. Generalment es defineix un cos com un anell amb element neutre multiplicatiu (A, +, ⋅, 0, 1) tal que tot element no nul és unitat, és a dir, tal que A* = A ∖ {0}. Alguns autors, però, ho anomenen i prefereixen reservar el terme cos per aquells casos on, a més a més, el producte és commutatiu.</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ar" >الوحدة (بالإنجليزية: Unit) أو العنصر القابل للعكس هي عنصر في حلقة يمتلك معكوسًا ضربيًّا. إذا كان عددًا صحيحًا جبريًّا يقسم كل عدد صحيح جبري في الحقل، تُسمَّى حينها وحدة في ذاك الحقل. يمكن لحقل ما أن يحتوي عددًا غير منتهٍ من الوحدات. وحدات هي العناصر الأولية فيما بينها وبين . وتسمى الوحدات المربعة الكاملة في . كل الحقول التربيعية الحقيقية تمتلك الوحدتين . أعداد الوحدات في الحقل التربيعي التخيلي حيث هي 4, 2, 6, 4, 2, 2, 2, 2, 4, 2, 2, 6, 2, ... (طالع هذه المتتالة من موسوعة المتتاليات الصحيحة على الإنترنت). ويوجد أربع وحدات لكل (طالع هذه المتتالية، وهي الأعداد المربعة الكاملة)، وستٌ لكل (طالع هذه المتتالية، وهي الأعداد المربعة الكاملة مضروبةً في ثلاثة)، واثنتان لكل الحقول التربيعية التخيلية الأخرى، أي (طالع هذه المتتالية). الجدول التالي يوضح الوحدات لقيم صغيرة لـ ، وتشير إلى الجذر المكعب للوحدة.</span><small> (ar)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="cs" >Jednotka neboli invertibilní prvek je v teorii okruhů takový prvek u nějakého okruhu R, pro který v daném okruhu existuje inverzní prvek, tedy prvek v splňující uv = vu = 1R, kde symbol 1R představuje jednotkový prvek (dolní index R označuje okruh — neutrální prvky různých okruhů se mohou vzájemně lišit).</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="eo" >En matematiko, inversigebla elemento en ringo estas elemento, kiu havas multiplikan inverson. La sama nocio (kaj terminon) estas aplikebla pli ĝenerale, por ĉia monoido. Tiam, ĉar la multiplika duongrupo de unuohava ringo estas monoido, inversigeblan elementon de ringo eblas difini kiel inversigeblan elementon de ĝia multiplika monoido.</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="de" >In der Algebra, einem Teilgebiet der Mathematik, wird ein invertierbares Element eines Monoids als Einheit bezeichnet. Einheiten werden vor allem in unitären Ringen betrachtet.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="es" >En matemática, especialmente en álgebra abstracta, el término unidad, elemento invertible o simplemente invertible en un anillo R con identidad multiplicativa 1R, se refiere a un elemento u tal que existe un v, llamado el inverso multiplicativo en R con u·v = v·u = 1R. Donde la operación · es la operación multiplicativa del anillo R. Elementos de esta naturaleza cumplen * El inverso multiplicativo es único * El conjunto de todos los invertibles junto con la operación multiplicativa del anillo forman un grupo denotado por U(R). Algo a tener en cuenta es que el término unidad debe diferenciarse de la 'unidad' en los anillos unitarios.</span><small> (es)</small></span></li> <li><span class="literal"><span property="dbo:abstract" lang="en" >In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit). Less commonly, the term unit is sometimes used to refer to the element 1 of the ring, in expressions like ring with a unit or unit ring, and also unit matrix. Because of this ambiguity, 1 is more commonly called the "unity" or the "identity" of the ring, and the phrases "ring with unity" or a "ring with identity" may be used to emphasize that one is considering a ring instead of a rng.</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ko" >추상대수학에서 가역원(可逆元, 영어: invertible element 또는 unit 유닛[*])은 환 또는 모노이드에서 곱셈에 대한 역원이 있는 원소들이다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="nl" >In de algebra, een deelgebied van de wiskunde heet een element van een unitaire ring , d.w.z. een (niet noodzakelijk commutatieve) ring met een neutraal element 1 voor de vermenigvuldiging, een eenheid in , als een invers element voor de vermenigvuldiging heeft. Eenvoudig geformuleerd: een eenheid is een deler van 1.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ja" >数学、とくに代数学における可逆元（かぎゃくげん、英: invertible element）または単元（たんげん、英: unit）とは、一般に代数系の乗法と呼ばれる二項演算に対する逆元を持つ元のことをいう。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="pt" >Em matemática, um elemento inversível ou uma unidade em um anel (unital) R refere-se a qualquer elemento u que tem seu elemento inverso no monoide multiplicativo de R, i.e. um elemento v que uv = vu = 1R, onde 1R é o elemento identidade multiplicativo. Infelizmente, o termo unidade é também usado referindo-se ao elemento identidade 1R do anel, em expressões como anel com uma unidade ou anel unidade, e também e.g. matriz 'unidade'. (Por esta razão, alguns autores chamam 1R "unidade", e dizem que R é um "anel com unidade" em vez de "anel com uma unidade".) Se no anel, então não é uma unidade. Se e a soma de qualquer duas não unidades não é uma unidade, então o anel é um anel local.</span><small> (pt)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="ru" >Обратимый элемент — элемент кольца с единицей, для которого существует обратный элемент относительно умножения. Другое название — делитель единицы. Также, в основном в переводах с английского, встречается название единица, что может вызывать путаницу с единичным элементом (в английских источниках используются два разных термина: unit element и Identity element). Иначе говоря, элемент кольца называется обратимым, если существует элемент , такой что где — единичный элемент кольца. Множество всех обратимых элементов кольца образует мультипликативную группу, называемую группой обратимых элементов (реже группой единиц). Эта группа всегда непустая, так как содержит как минимум единицу кольца.</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="pl" >Element odwracalny – dla danego (wewnętrznego) działania dwuargumentowego określonego w pewnej strukturze algebraicznej element, dla którego istnieje element do niego odwrotny względem tego działania. Innymi słowy, jeżeli zbiór wyposażony jest w działanie to element jest odwracalny, jeśli istnieje taki element dla którego spełnione są równości oraz gdzie jest elementem neutralnym działania Jeżeli spełniony jest tylko pierwszy warunek, to element nazywa się prawostronnie odwracalnym, jeżeli wyłącznie drugi, to nazywa się go lewostronnie odwracalnym. Łączność działania gwarantuje, że elementy odwracalne jednostronnie są odwracalne obustronnie, z kolei przemienność tego działania sprawia, że elementy tak lewo- i, jak i prawostronnie odwracalne są odwracalne obustronnie.</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="uk" >Оборотний елемент, також одиниця кільця чи дільник одиниці — будь-який елемент кільця, для якого існує обернений елемент, тобто є такий елемент , що . Множина всіх О. е. (одиниць кільця) утворює мультиплікативну групу, яку називають групою одиниць або групою О. е.. Якщо — дільник одиниці, тоді елементи і називаються асоційованими з . Зазвичай поняття дільника одиниці й асоційованого елемента вживається для областей цілісності.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="dbo:abstract" lang="zh" >单位又被称为可逆元。在數學裡，於一(有单位的)環 內的可逆元是指一 的可逆元素，即一元素 使得存在一於 內的 有下列性質：，其中 是乘法單位元。 亦即， 是 內乘法幺半群的一可逆元素。</span><small> (zh)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageID"><small>dbo:</small>wikiPageID</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><span property="dbo:wikiPageID" datatype="xsd:integer" >289666</span><small> (xsd:integer)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://dbpedia.org/ontology/wikiPageLength"><small>dbo:</small>wikiPageLength</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><span property="dbo:wikiPageLength" datatype="xsd:nonNegativeInteger" >11077</span><small> 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href="http://www.w3.org/2002/07/owl#Thing"><small>owl</small>:Thing</a></span></li> <li><span class="literal"><a class="uri" rel="rdf:type" resource="http://dbpedia.org/ontology/Person" href="http://dbpedia.org/ontology/Person"><small>dbo</small>:Person</a></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#comment"><small>rdfs:</small>comment</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="cs" >Jednotka neboli invertibilní prvek je v teorii okruhů takový prvek u nějakého okruhu R, pro který v daném okruhu existuje inverzní prvek, tedy prvek v splňující uv = vu = 1R, kde symbol 1R představuje jednotkový prvek (dolní index R označuje okruh — neutrální prvky různých okruhů se mohou vzájemně lišit).</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="eo" >En matematiko, inversigebla elemento en ringo estas elemento, kiu havas multiplikan inverson. La sama nocio (kaj terminon) estas aplikebla pli ĝenerale, por ĉia monoido. Tiam, ĉar la multiplika duongrupo de unuohava ringo estas monoido, inversigeblan elementon de ringo eblas difini kiel inversigeblan elementon de ĝia multiplika monoido.</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="de" >In der Algebra, einem Teilgebiet der Mathematik, wird ein invertierbares Element eines Monoids als Einheit bezeichnet. Einheiten werden vor allem in unitären Ringen betrachtet.</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ko" >추상대수학에서 가역원(可逆元, 영어: invertible element 또는 unit 유닛[*])은 환 또는 모노이드에서 곱셈에 대한 역원이 있는 원소들이다.</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="nl" >In de algebra, een deelgebied van de wiskunde heet een element van een unitaire ring , d.w.z. een (niet noodzakelijk commutatieve) ring met een neutraal element 1 voor de vermenigvuldiging, een eenheid in , als een invers element voor de vermenigvuldiging heeft. Eenvoudig geformuleerd: een eenheid is een deler van 1.</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ja" >数学、とくに代数学における可逆元（かぎゃくげん、英: invertible element）または単元（たんげん、英: unit）とは、一般に代数系の乗法と呼ばれる二項演算に対する逆元を持つ元のことをいう。</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="uk" >Оборотний елемент, також одиниця кільця чи дільник одиниці — будь-який елемент кільця, для якого існує обернений елемент, тобто є такий елемент , що . Множина всіх О. е. (одиниць кільця) утворює мультиплікативну групу, яку називають групою одиниць або групою О. е.. Якщо — дільник одиниці, тоді елементи і називаються асоційованими з . Зазвичай поняття дільника одиниці й асоційованого елемента вживається для областей цілісності.</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="zh" >单位又被称为可逆元。在數學裡，於一(有单位的)環 內的可逆元是指一 的可逆元素，即一元素 使得存在一於 內的 有下列性質：，其中 是乘法單位元。 亦即， 是 內乘法幺半群的一可逆元素。</span><small> (zh)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ar" >الوحدة (بالإنجليزية: Unit) أو العنصر القابل للعكس هي عنصر في حلقة يمتلك معكوسًا ضربيًّا. إذا كان عددًا صحيحًا جبريًّا يقسم كل عدد صحيح جبري في الحقل، تُسمَّى حينها وحدة في ذاك الحقل. يمكن لحقل ما أن يحتوي عددًا غير منتهٍ من الوحدات. وحدات هي العناصر الأولية فيما بينها وبين . وتسمى الوحدات المربعة الكاملة في .</span><small> (ar)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ca" >En matemàtiques, un element invertible d'un conjunt amb una és aquell del qual es pot obtenir un element invers per aquesta llei. Si la llei és associativa, els elements invertibles del conjunt tenen cadascun un únic invers. En teoria d'anells, els elements invertibles per la segona llei de composició interna (o producte) d'un anell s'acostumen a anomenar unitats. Si tenim un anell amb element neutre (A, +, ⋅, 0, 1), es denota A* el conjunt de les unitats, que és un grup amb la multiplicació i s'anomena grup de les unitats.</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="es" >En matemática, especialmente en álgebra abstracta, el término unidad, elemento invertible o simplemente invertible en un anillo R con identidad multiplicativa 1R, se refiere a un elemento u tal que existe un v, llamado el inverso multiplicativo en R con u·v = v·u = 1R. Donde la operación · es la operación multiplicativa del anillo R. Elementos de esta naturaleza cumplen * El inverso multiplicativo es único * El conjunto de todos los invertibles junto con la operación multiplicativa del anillo forman un grupo denotado por U(R).</span><small> (es)</small></span></li> <li><span class="literal"><span property="rdfs:comment" lang="en" >In algebra, a unit of a ring is an invertible element for the multiplication of the ring. That is, an element u of a ring R is a unit if there exists v in R such that where 1 is the multiplicative identity; the element v is unique for this property and is called the multiplicative inverse of u. The set of units of R forms a group R× under multiplication, called the group of units or unit group of R. Other notations for the unit group are R∗, U(R), and E(R) (from the German term Einheit).</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="pl" >Element odwracalny – dla danego (wewnętrznego) działania dwuargumentowego określonego w pewnej strukturze algebraicznej element, dla którego istnieje element do niego odwrotny względem tego działania. Innymi słowy, jeżeli zbiór wyposażony jest w działanie to element jest odwracalny, jeśli istnieje taki element dla którego spełnione są równości oraz gdzie jest elementem neutralnym działania</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="pt" >Em matemática, um elemento inversível ou uma unidade em um anel (unital) R refere-se a qualquer elemento u que tem seu elemento inverso no monoide multiplicativo de R, i.e. um elemento v que uv = vu = 1R, onde 1R é o elemento identidade multiplicativo. Infelizmente, o termo unidade é também usado referindo-se ao elemento identidade 1R do anel, em expressões como anel com uma unidade ou anel unidade, e também e.g. matriz 'unidade'. (Por esta razão, alguns autores chamam 1R "unidade", e dizem que R é um "anel com unidade" em vez de "anel com uma unidade".)</span><small> (pt)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:comment" lang="ru" >Обратимый элемент — элемент кольца с единицей, для которого существует обратный элемент относительно умножения. Другое название — делитель единицы. Также, в основном в переводах с английского, встречается название единица, что может вызывать путаницу с единичным элементом (в английских источниках используются два разных термина: unit element и Identity element). Иначе говоря, элемент кольца называется обратимым, если существует элемент , такой что где — единичный элемент кольца.</span><small> (ru)</small></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" href="http://www.w3.org/2000/01/rdf-schema#label"><small>rdfs:</small>label</a> </td><td class="col-10 text-break"><ul> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ar" >وحدة (نظرية الحلقات)</span><small> (ar)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ca" >Element invertible</span><small> (ca)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="cs" >Jednotka (teorie okruhů)</span><small> (cs)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="de" >Einheit (Mathematik)</span><small> (de)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="eo" >Inversigebla elemento</span><small> (eo)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="es" >Unidad (álgebra)</span><small> (es)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ko" >가역원</span><small> (ko)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ja" >可逆元</span><small> (ja)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="pl" >Element odwracalny</span><small> (pl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="nl" >Eenheid (algebra)</span><small> (nl)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="pt" >Unidade (teoria dos anéis)</span><small> (pt)</small></span></li> <li><span class="literal"><span property="rdfs:label" lang="en" >Unit (ring theory)</span><small> (en)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="ru" >Обратимый элемент</span><small> (ru)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="uk" >Оборотний елемент</span><small> (uk)</small></span></li> <li style="display:none;"><span class="literal"><span property="rdfs:label" lang="zh" >可逆元</span><small> (zh)</small></span></li> </ul></td></tr><tr class="even"><td class="col-2"><a class="uri" href="http://www.w3.org/2002/07/owl#differentFrom"><small>owl:</small>differentFrom</a> </td><td class="col-10 text-break"><ul> <li><span class="literal"><a class="uri" rel="owl:differentFrom" resource="http://dbpedia.org/resource/Unit_ring" href="http://dbpedia.org/resource/Unit_ring"><small>dbr</small>:Unit_ring</a></span></li> </ul></td></tr><tr class="odd"><td class="col-2"><a class="uri" 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