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.

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.

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

* 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.

* Suppose G and H are topologically finitely-generated profinite groups which are isomorphic as discrete groups by an isomorphism ι.

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.

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.

* Every finitely-generated commutative algebra over a commutative Noetherian ring is Noetherian.

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.

As a consequence, any finitely-generated module over an Artinian ring is Artinian.

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.

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

Statement 1: Let I be an ideal in R, and M a finitely-generated module over R. If IM

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.

This coincides with the previous example when M is a finitely-generated, free R-module.

This generalizes an earlier analogous result of Jean-Pierre Serre for topologically finitely-generated pro-p groups.

R will be a local ring with maximal ideal m < sub > R </ sub >, and M and N are finitely-generated R-modules.

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.

Let ( R, m ) be a Noetherian local ring with maximal ideal m, and let M be a finitely-generated R-module.

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.

groups ( G, *) and ( H, ●), denoted by G × H. For abelian groups which are written additively, it may also be called the direct sum of two groups, denoted by.

While the description of strong and weak interactions within the Standard Model is based on gauge symmetries governed by the simple symmetry groups SU ( 3 ) and SU ( 2 ) which allow only discrete charges, the remaining component, the weak hypercharge interaction is described by an abelian symmetry U ( 1 ) which in principle allows for arbitrary charge assignments.

This applies to the I-indexed diagrams in the category of R-modules, with R a commutative ring ; it is not necessarily true in an arbitrary abelian category ( see Roos ' " Derived functors of inverse limits revisited " for examples of abelian categories in which lim ^ n, on diagrams indexed by a countable set, is nonzero for n > 1 ).

Any one-dimensional Lie algebra over a field is abelian, by the antisymmetry of the Lie bracket.

More precisely: if A is an abelian group and we denote by m the group multiplication of A, by e the inclusion of the identity element, and by inv the inversion operation on A, then ( A, m, e, inv ) is a group object in the category of groups ( or monoids ).

In the category of abelian groups, biproducts always exist and are given by the direct sum.

In abstract algebra, the endomorphism ring of an abelian group X, denoted by End ( X ), is the set of all homomorphisms of X into itself.

They are called addition and multiplication and commonly denoted by "+" and "⋅"; e. g. a + b and a ⋅ b. To form a ring these two operations have to satisfy a number of properties: the ring has to be an abelian group under addition as well as a monoid under multiplication, where multiplication distributes over addition ; i. e., a ⋅ ( b + c )

Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom ( A, B ), which is an abelian group by item 1 ; or as the unique morphism A → O → B, where O is a zero object, guaranteed to exist by item 2.

Here 0 denotes the trivial abelian group with a single element, the map from Z to Z is multiplication by 2, and the map from Z to the factor group Z / 2Z is given by reducing integers modulo 2.

A chain complex is a sequence of abelian groups or modules ... A < sub > 2 </ sub >, A < sub > 1 </ sub >, A < sub > 0 </ sub >, A < sub >- 1 </ sub >, A < sub >- 2 </ sub >, ... connected by homomorphisms ( called boundary operators ) d < sub > n </ sub >: A < sub > n </ sub >→ A < sub > n − 1 </ sub >, such that the composition of any two consecutive maps is zero: d < sub > n </ sub > ∘ d < sub > n + 1 </ sub > = 0 for all n. They are usually written out as:

A cochain complex is a sequence of abelian groups or modules ...,,,,,, ... connected by homomorphisms such that the composition of any two consecutive maps is zero: for all n:

Define C < sub > n </ sub >( X ) for natural n to be the free abelian group formally generated by singular n-simplices in X, and define the boundary map

* ' Tohoku ': The approach in a celebrated paper by Alexander Grothendieck which appeared in the Second Series of the Tohoku Mathematical Journal in 1957, using the abelian category concept ( to include sheaves of abelian groups ).

A chain complex is a sequence of abelian groups or modules C < sub > 0 </ sub >, C < sub > 1 </ sub >, C < sub > 2 </ sub >, ... connected by homomorphisms which are called boundary operators.

The simplicial homology groups H < sub > n </ sub >( X ) of a simplicial complex X are defined using the simplicial chain complex C ( X ), with C ( X )< sub > n </ sub > the free abelian group generated by the n-simplices of X.