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A finitely-generated abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of ( finitely many ) indecomposable abelian groups.

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## Some Related Sentences

finitely-generated and abelian

The structural result

**of**Mordell, that**the**rational points on an elliptic curve E**form****a**__finitely-generated____abelian__**group**, used an infinite descent argument based on E / 2E in Fermat's style**.**
Most

**of**these can be posed**for**an__abelian__variety**A**over**a****number**field K ;**or**more generally**(****for**global fields**or**more general__finitely-generated__rings**or**fields ).
The basic result

**(**Mordell – Weil theorem**)**says that**A****(**K ),**the****group****of**points on**A**over K,**is****a**__finitely-generated____abelian__**group****.**
As

**a****group****(**dropping its geometric structure**)****a**lattice**is****a**__finitely-generated__free__abelian__**group****.**
The

__abelian__**group****of**divisor classes up**to**algebraic equivalence**is**now called**the**Néron-Severi**group**;**it****is**known**to**be**a**__finitely-generated____abelian__**group**,**and****the**result**is**about its tensor product with**the**rational**number**field**.**
The quotient Pic

**(**V )/ Pic < sup > 0 </ sup >( V**)****is****a**__finitely-generated____abelian__**group**denoted NS**(**V ),**the**Néron – Severi**group****of**V**.**In other words**the**Picard**group**fits into an exact sequence**a**

__finitely-generated__

__abelian__

**group**by

**the**Néron – Severi theorem, which was proved by Severi over

**the**complex numbers

**and**by Néron over more general fields

**.**

finitely-generated and group

* According

**to****a**theorem**of**Nikolay Nikolov**and**Dan Segal, in any topologically__finitely-generated__profinite__group__**(**that**is**,**a**profinite__group__that has**a**dense__finitely-generated__subgroup**)****the**subgroups**of**finite index are open**.**
* As an easy corollary

**of****the**Nikolov-Segal result above, any surjective discrete__group__homomorphism φ: G → H between profinite**groups**G**and**H**is**continuous as long as G**is**topologically__finitely-generated__**.**
Therefore

**the**topology on**a**topologically__finitely-generated__profinite__group__**is**uniquely determined by its algebraic structure**.****)**The usual terminology

**is**different:

**a**

__group__G

**is**called locally finite

**if**every

__finitely-generated__subgroup

**is**finite

**.**

finitely-generated and is

If R

__is__complete, then there exists**a**__finitely-generated__R-module M ≠ 0 such that**some****(**equivalently every**)**system**of**parameters**for**R__is__**a**regular sequence on M**.**
In fact,

**it**__is__**a**__finitely-generated__module over its center ; even more so,**it**__is__an Azumaya algebra over its center**.**
When R

__is__Noetherian**and**M__is__**a**__finitely-generated__R-module, being flat__is__**the**same as being locally free in**the**following sense: M__is__**a**flat R-module**if****and****only****if****for**every**prime**ideal**(****or**even just**for**every maximal ideal**)**P**of**R,**the**localization__is__free as**a**module over**the**localization**.**
Daniel Lazard proved in 1969 that

**a**module M__is__flat**if****and****only****if****it**__is__**a****direct**limit**of**__finitely-generated__free modules**.**

finitely-generated and if

Since an Artinian ring

**is**also Noetherian,**and**__finitely-generated__modules over**a**Noetherian ring are Noetherian,**it****is**true that**for**an Artinian ring R, any__finitely-generated__R-module**is**both Noetherian**and**Artinian,**and****is**said**to**be**of**finite length ; however,__if__R**is**not Artinian,**or**__if__M**is**not**finitely**generated, there are counterexamples**.**

finitely-generated and isomorphic

* Suppose G

**and**H are topologically__finitely-generated__profinite**groups**which are__isomorphic__as discrete**groups**by an isomorphism ι**.**

finitely-generated and Z

In

**the**case**of****a**finite simplicial complex**the**homology**groups**H < sub > k </ sub >( X,__Z__**)**are__finitely-generated__,**and**so has**a**finite rank**.**

finitely-generated and form

If S has

**the**additional property that**is****a**coherent sheaf**and**locally generates S over**(**that**is**, when we pass**to****the**stalk**of****the**sheaf S at**a**point x**of**X, which**is****a**graded algebra whose degree-zero elements__form__**the**ring then**the**degree-one elements__form__**a**__finitely-generated__module over**and**also generate**the**stalk as an algebra over**it****)**then we may make**a**further construction**.**

finitely-generated and for

This generalizes an earlier analogous result

**of**Jean-Pierre Serre__for__topologically__finitely-generated__pro-p**groups****.**

finitely-generated and .

R will be

**a**local ring with maximal ideal m < sub > R </ sub >,**and**M**and**N are__finitely-generated__R-modules__.__
Let

**(**R, m**)**be**a**Noetherian local ring with maximal ideal m,**and**let M be**a**__finitely-generated__R-module__.__

finitely-generated and Every

abelian and group

An associative R-algebra

**is**an additive__abelian____group__**A**which has**the**structure**of**both**a**ring**and**an R-module in such**a**way that ring multiplication**is**R-bilinear:
In abstract algebra, an

__abelian____group__, also called**a**commutative__group__,**is****a**__group__in which**the**result**of**applying**the**__group__operation**to**two__group__elements does not depend on their order**(****the**axiom**of**commutativity ).
The concept

**of**an__abelian____group__**is**one**of****the**first concepts encountered in undergraduate abstract algebra, with**many**other basic objects, such as**a**module**and****a**vector space, being its refinements**.**
An

__abelian____group__**is****a**set,**A**, together with an operation "•" that combines any two elements**a****and**b**to****form**another element denoted**.**
To qualify as an

__abelian____group__,**the**set**and**operation,, must satisfy five requirements known as**the**__abelian____group__axioms:
For

**the**case**of****a**non-commutative base ring R**and****a**right module M < sub > R </ sub >**and****a**left module < sub > R </ sub > N, we can define**a**bilinear map, where T**is**an__abelian____group__, such that**for**any**n**in N,**is****a**__group__homomorphism,**and****for**any m in M,**is****a**__group__homomorphism too,**and**which also satisfies
Their classification

**is**divided into**the**small**and**large rank cases, where**the**rank**is****the**largest rank**of**an odd__abelian__subgroup normalizing**a**nontrivial 2-subgroup, which**is**often**(**but not always**)****the**same as**the**rank**of****a**Cartan subalgebra when**the**__group__**is****a**__group__**of**Lie type in characteristic 2**.**
The commutator subgroup

**is**important because**it****is****the**smallest normal subgroup such that**the**quotient__group__**of****the**original__group__by this subgroup**is**__abelian__**.**
So in

**some**sense**it**provides**a**measure**of**how far**the**__group__**is**from being__abelian__;**the**larger**the**commutator subgroup**is**,**the**" less__abelian__"**the**__group__**is****.**

abelian and is

The theory

**of**__abelian__**groups**__is__generally simpler than that**of**their non-abelian counterparts,**and**finite__abelian__**groups**are very well understood**.**0.151 seconds.