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Finitely generated module

In mathematics, a finitely generated module is a module that has a finite generating set. A finitely generated module over a ring R may also be called a finite R-module, finite over R, or a module of finite type.

Definition, Equivalent definitions and finitely cogenerated modules & Finitely generated modules over a commutative ring

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Definition

SetR-linear mapQuotientFree moduleCardinality
14 related topics

Equivalent definitions and finitely cogenerated modules

SubsetChainEpimorphismMorita equivalenceDual
13 related topics

Finitely generated modules over a commutative ring

Nakayama's lemmaSurjectiveInjectiveAutomorphismHopfian module
12 related topics

Textbooks

Atiyah, M. F.Macdonald, I. G.Introduction to Commutative AlgebraMRBourbaki, Nicolas
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Overview

Definition

Examples

Some facts

Finitely generated modules over a commutative ring

Generic rank

Equivalent definitions and finitely cogenerated modules

Finitely presented, finitely related, and coherent modules

Textbooks

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Finitely generated module

Nodes104
Edges103
Triples54
Avg. degree1.98
Density0.019231
Components1

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Finitely generated module

Top relations

related to Generic rank · 15
Finitely generated module → A-linearly, By, Hilbert, Let, M/F, Noetherian, Now, Poincaré, Rank, Serre, Since, Suppose, Then, This, When
related to Finitely generated modules over a commutative ring · 14
Finitely generated module → Any R-module, Artinian, For, Hopfian, Nakayama's, Noetherian, R-endomorphism, R-submodules, Similarly, Sometimes, Swan, The Forster, This, Tor
related to Some facts · 12
Finitely generated module → As, Both, Consider, Every, Hilbert's, In, Noetherian, R-module, Since, This, X1, X2
related to Examples · 8
Finitely generated module → Finitely, If, Indeed, Let, Noetherian, R-submodule, Then, These
related to Finitely presented, finitely related, and coherent modules · 3
Finitely generated module → Another, Rk, Suppose
is a · 2
Finitely generated module → finitely generated algebra, module that has a finite generating set

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Important terminology

finitely module generated ring modules finite noetherian displaystyle presented submodules set coherent commutative also free submodule algebra rank every integral

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Finitely generated moduleis amodule that has a finite generating set0.90text
Finitely generated moduleis afinitely generated algebra0.90text
Finitely generated modulerelated to ExamplesIf0.60section
Finitely generated modulerelated to ExamplesLet0.60section
Finitely generated modulerelated to ExamplesThen0.60section
Finitely generated modulerelated to ExamplesR-submodule0.60section
Finitely generated modulerelated to ExamplesIndeed0.60section
Finitely generated modulerelated to ExamplesNoetherian0.60section
Finitely generated modulerelated to ExamplesFinitely0.60section
Finitely generated modulerelated to ExamplesThese0.60section
Finitely generated modulerelated to Finitely generated modules over a commutative ringFor0.60section
Finitely generated modulerelated to Finitely generated modules over a commutative ringNakayama's0.60section

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