About: Structure theorem for finitely generated modules over a principal ideal domain

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In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields.

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rdf:type	yago:WikicatMatrices yago:WikicatTheoremsInAbstractAlgebra yago:Abstraction100002137 yago:Arrangement107938773 yago:Array107939382 yago:Communication100033020 yago:Group100031264 yago:Matrix108267640 yago:Message106598915 yago:Proposition106750804 yago:Statement106722453 yago:Theorem106752293
rdfs:label	Teorema d'estructura dels mòduls finitament generats sobre un domini d'ideals principals (ca) Théorème des facteurs invariants (fr) 主イデアル整域上の有限生成加群の構造定理 (ja) Structure theorem for finitely generated modules over a principal ideal domain (en) Структурная теорема для конечнопорождённых модулей над областями главных идеалов (ru) Структурна теорема для скінченнопороджених модулів над областями головних ідеалів (uk)
rdfs:comment	En matemàtiques, en l'àmbit de l'àlgebra abstracta, el teorema d'estructura dels mòduls finitament generats sobre un domini d'ideals principals és una generalització del i, expressat d'una manera informal, estableix que els mòduls finitament generats es poden descompondre unívocament, de forma similar a la factorització en nombres primers dels enters. Aquest resultat proporciona un marc de treball senzill per entendre diversos enunciats sobre formes canòniques per matrius quadrades definides sobre un cos. (ca) In mathematics, in the field of abstract algebra, the structure theorem for finitely generated modules over a principal ideal domain is a generalization of the fundamental theorem of finitely generated abelian groups and roughly states that finitely generated modules over a principal ideal domain (PID) can be uniquely decomposed in much the same way that integers have a prime factorization. The result provides a simple framework to understand various canonical form results for square matrices over fields. (en) 数学において，抽象代数学の分野において，主イデアル整域上の有限生成加群の構造定理は有限生成アーベル群の基本定理の一般化であり，あらっぽく言えば，有限生成加群は整数の素因数分解とほぼ同じように一意的に分解するというものである．この結果は体上の正方行列に対する様々な標準形の結果を理解する単純な枠組みを提供する． (ja) Структурная теорема для конечнопорождённых модулей над областями главных идеалов является обобщением теоремы о классификации конечнопорождённых абелевых групп. Эта теорема предоставляет общий способ понимания некоторых результатов о канонических формах матриц. (ru) Структурна теорема для скінченнопороджених модулів над областями головних ідеалів є узагальненням теореми про класифікацію скінченнопороджених абелевих груп. Ця теорема надає загальний спосіб розуміння деяких результатів про канонічні форми матриць. (uk) En mathématiques, le théorème des facteurs invariants porte sur les modules de type fini sur les anneaux principaux. Les facteurs invariants non inversibles sont des obstructions à l'inversibilité des matrices qui n'apparaissent pas dans la théorie des espaces vectoriels. Leur calcul a de nombreuses applications : par exemple trouver la classe d'isomorphie d'un groupe abélien de type fini à partir d'une présentation de celui-ci. Dans un cadre précis, le théorème des facteurs invariants se particularise en théorèmes de réduction d'endomorphisme. Il permet alors notamment de calculer les invariants de similitude d'un endomorphisme sur un espace vectoriel. Il joue un rôle essentiel dans la résolution de systèmes d'équations différentielles linéaires à coefficients constants et dans la théorie (fr)
dcterms:subject	Module theory Theorems in abstract algebra
Wikipage page ID	9902787 (xsd:integer)
Wikipage revision ID	1014312172 (xsd:integer)
Link from a Wikipage to another Wikipage	Principal ideal domain Module (mathematics) Torsion submodule Algebra over a field Algebraically closed field Module theory Jordan block Jordan–Hölder theorem Characteristic polynomial Cyclic module Unit (ring theory) Vector space Dedekind domain Indecomposable module Invariant factor Companion matrix Mathematics Noetherian ring Quotient module Frobenius normal form Fundamental theorem of finitely generated abelian groups Complete set of invariants Composition series Embedding Ideal (ring theory) Ideal class group Kernel (algebra) Krull–Schmidt theorem Automorphism Large cardinal Linear subspace Minimal polynomial (field theory) Square matrix Field (mathematics) Finite set Finitely generated module Direct sum of modules Primary decomposition Finitely presented module Primary ideal Prime element Projective module Rational canonical form Isomorphic Baer–Specker group Theorems in abstract algebra Abstract algebra Coherent ring Torsion-free module Polynomial Free module Integer Canonical form Set theory Smith normal form Lasker-Noether theorem Lasker–Noether theorem Invariant factors Jordan canonical form Basis (vector space) Steinitz class Prime factorization Primary rational canonical form
sameAs	Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain Structure theorem for finitely generated modules over a principal ideal domain

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