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finitely-generated and abelian
He also introduced the structure theorem for finitely-generated abelian groups.
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.
* Every finitely-generated locally cyclic group is cyclic.

finitely-generated and is
This coincides with the previous example when M is a finitely-generated, free R-module.
* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.
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.
As a consequence, any finitely-generated module over an Artinian ring is Artinian.

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.
so every finitely-generated module over a PID is completely decomposable.

finitely-generated and Every
Every finitely-generated R-module is a direct sum of these.

abelian and group
Generally speaking, negation is an automorphism of any abelian group, but not of a ring or field.
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:
More compactly, an abelian group is a commutative group.
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
His notion of abelian category is now the basic object of study in homological algebra.
The theory of abelian groups is generally simpler than that of their non-abelian counterparts, and finite abelian groups are very well understood.
On the other hand, the theory of infinite abelian groups is an area of current research.
In other words, G / N is abelian if and only if N contains the commutator subgroup.

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