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In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces.

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rdf:type	software
rdfs:label	Länge (Algebra) (de) Longueur d'un module (fr) Lunghezza di un modulo (it) Length of a module (en) 가군의 길이 (ko) 加群の長さ (ja) Длина модуля (ru) Довжина модуля (uk) 長度 (模論) (zh)
rdfs:comment	Im mathematischen Teilgebiet der Algebra ist die Länge ein Maß für die Größe eines Moduls. (de) La longueur d'un module M sur un anneau A est un entier naturel ou l'infini. Elle généralise d'une certaine manière la notion de dimension d'un espace vectoriel sur un corps. Les modules de longueur finie ont beaucoup de particularités généralisant celles des espaces vectoriels de dimension finie. (fr) In matematica, la lunghezza di un modulo è una quantità (un numero naturale oppure infinito) che misura la "grandezza" di un modulo, generalizzando la nozione di dimensione degli spazi vettoriali. (it) 환론에서, 가군의 길이(영어: length)는 가군의 크기를 나타내는 측도이며, 벡터 공간의 차원의 일반화이다. (ko) 抽象代数学において、加群の長さ (length) は加群の「大きさ」の尺度である。それは部分加群の最長の鎖の長さと定義され、ベクトル空間の次元の概念の一般化である。有限の長さをもつ加群は有限次元ベクトル空間と多くの重要な性質を共有する。環と加群の理論において「大きさを測る」ために使われる他の概念は深さと高さである。これらは両方とも定義するのが幾分デリケートである。これらはまた有用な次元のさまざまなアイデアである。長さ有限の可換環は形式的な代数幾何学の関手的扱いにおいて本質的な役割を果たす。 (ja) Длина модуля — способ измерения «размера» модуля, обобщающий понятие размерности векторного пространства. Длина определяется как максимальная длина цепочки вложенных подмодулей. (ru) В абстрактній алгебрі довжина модуля — числова характеристика модуля, що деякою мірою узагальнює поняття розмірності векторного простору. (uk) 在數學中，設為環，一個 -模之長度是一個整數（包括無窮大），它推廣了向量空間的維度。有限長度的模與有限維向量空間有許多共通性。 (zh) In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces. (en)
dcterms:subject	Module theory
Wikipage page ID	816070 (xsd:integer)
Wikipage revision ID	1092163955 (xsd:integer)
Link from a Wikipage to another Wikipage	Projective space Module (mathematics) Multiplicity (mathematics) Meromorphic function Serre's multiplicity conjectures Deformation theory Module theory Unit (ring theory) Vector space Complex analysis Chow ring Module theory Short exact sequence Commutative ring Composition series Krull–Schmidt theorem Bézout's theorem Weierstrass factorization theorem Weil divisor Local ring Affine variety Algebraic variety Cyclic group Finitely generated module Dimension (vector space) Hilbert–Poincaré series Rational function Ring (mathematics) Ring theory Height (ring theory) Hilbert scheme Intersection theory Jean-Pierre Serre Hypersurface Prime number Artin ring Abstract algebra Chinese remainder theorem Codimension Dimension Dimension theory (algebra) Artinian module Integer Scheme (mathematics) Noetherian module Simple module Intersection multiplicity Submodule Artinian local ring Formal algebraic geometry Projective surface Depth (algebra) Hopkins' theorem Stalk of a sheaf Subvariety (mathematics)
Link from a Wikipage to an external page	http://www.centerofmathematics.com/wwcomstore/index.php/commalg.html https://stacks.math.columbia.edu/tag/00IU
sameAs	Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module Length of a module
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has abstract	Im mathematischen Teilgebiet der Algebra ist die Länge ein Maß für die Größe eines Moduls. (de) In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. page 153 In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules. Modules with finite length share many important properties with finite-dimensional vector spaces. Other concepts used to 'count' in ring and module theory are depth and height; these are both somewhat more subtle to define. Moreover, their use is more aligned with dimension theory whereas length is used to analyze finite modules. There are also various ideas of dimension that are useful. Finite length commutative rings play an essential role in functorial treatments of formal algebraic geometry and deformation theory where Artin rings are used extensively. (en) La longueur d'un module M sur un anneau A est un entier naturel ou l'infini. Elle généralise d'une certaine manière la notion de dimension d'un espace vectoriel sur un corps. Les modules de longueur finie ont beaucoup de particularités généralisant celles des espaces vectoriels de dimension finie. (fr)

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