About: Idempotent (ring theory)

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In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix.

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rdfs:label	Idempotentes Element (de) Idempotent (ring theory) (en) 冪等元 (ja) 멱등원 (ko) Idempotent element (nl) Ідемпотентний елемент (uk)
rdfs:comment	抽象代数学において、二項演算 ∗ をもった集合の元 x は x ∗ x = x であるときに冪等元（英: idempotent element）あるいは単に冪等（英: idempotent）と呼ばれる。これはその特定の元における二項演算の冪等性を反映している。環論において（積に関する）冪等元は特に重要である。一般の環に対して、冪等元は加群の分解や環のホモロジー的性質と深く関わっている。この概念は , pp. 16–17) によって導入された。本記事は環論的な意味の冪等元を扱う。 (ja) 환론과 모노이드 이론에서, 멱등원(蓂等元, 영어: idempotent element)은 거듭제곱하여도 변하지 않는 원소이다. (ko) Een idempotent element in een magma is een element waarvoor geldt: In een algebraïsche structuur met een neutraal element, zoals een groep, is het neutrale element idempotent. In een groep is het neutrale element ook het enige idempotente element. Een algebraïsche structuur kan meerdere idempotente elementen hebben. (nl) Ідемпотентний елемент (лат. idem — такий самий, potens — сильний) — в математиці, елемент півгрупи чи кільця, що є рівним своєму квадрату: . (uk) In ring theory, a branch of abstract algebra, an idempotent element or simply idempotent of a ring is an element a such that a2 = a. That is, the element is idempotent under the ring's multiplication. Inductively then, one can also conclude that a = a2 = a3 = a4 = ... = an for any positive integer n. For example, an idempotent element of a matrix ring is precisely an idempotent matrix. (en)
dcterms:subject	Ring theory
Wikipage page ID	361940 (xsd:integer)
Wikipage revision ID	1117765581 (xsd:integer)
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Link from a Wikipage to an external page	http://www.math.harvard.edu/history/peirce_algebra/index.html http://foldoc.org/idempotent
sameAs	Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory) Idempotent (ring theory)
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