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.

However, constructed an example of an abelian variety where Hdg < sup > 2 </ sup >( X ) is not generated by products of divisor classes.

A divisor D is an element of the free abelian group on the points of the surface.

* A module is called coprimary if every zero divisor of M is nilpotent in M. For example, groups of prime power order and free abelian groups are coprimary modules over the ring of integers.

The set H of congruence classes of 0, 4, and 8 modulo 12 is a subgroup of order 3, and it is a normal subgroup since any subgroup of an abelian group is normal.

Prior to this, topological classes in combinatorial topology were not formally considered as abelian groups.

However, if R is the ring of algebraic integers in an algebraic number field, or more generally a Dedekind domain, the multiplication defined above turns the set of fractional ideal classes into an abelian group, the ideal class group of R. The group property of existence of inverse elements follows easily from the fact that, in a Dedekind domain, every non-zero ideal ( except R ) is a product of prime ideals.

* The Shafarevich conjecture that there are only finitely many isomorphism classes of abelian varieties of fixed dimension and fixed polarization degree over a fixed number field with good reduction outside a given finite set of places ; and

The theory takes its name from the fact that it provides a one-to-one correspondence between finite abelian extensions of a fixed global field and appropriate classes of ideals of the field or open subgroups of the idele class group of the field.

Their use replaced the classes of ideals and essentially clarified and simplified structures which describe abelian extensions of global fields.

Two important classes of algebraic groups arise, that for the most part are studied separately: abelian varieties ( the ' projective ' theory ) and linear algebraic groups ( the ' affine ' theory ).

As stated above, using the Grothendieck additivity axiom for Chern classes, the first of these identities can be generalized to state that ch is a homomorphism of abelian groups from the K-theory K ( X ) into the rational cohomology of X.

is abelian, so conjugacy classes become elements ), then for p not dividing m the Frobenius class in the Galois group is p mod m. From this point of view,

If one requires G and Q to be abelian groups, then the set of isomorphism classes of extensions of Q by a given ( abelian ) group N is in fact a group, which is isomorphic to

The first Stiefel – Whitney class classifies smooth real line bundles ; in particular, the collection of ( equivalence classes of ) real line bundles are in correspondence with elements of the first cohomology with Z / 2Z coefficients ; this correspondence is in fact an isomorphism of abelian groups ( the group operations being tensor product of line bundles and the usual addition on cohomology ).

The change of name reflected the move to organise topological classes such as cycles modulo boundaries explicitly into abelian groups.

In the case of surfaces, the Enriques classification was into five similar big classes, with three of those being analogues of the curve cases, and two more ( elliptic fibrations, and K3 surfaces, as they would now be called ) being with the case of two-dimension abelian varieties in the ' middle ' territory.

In mathematics, the Brauer group of a field K is an abelian group whose elements are Morita equivalence classes of central simple algebras of finite rank over K and addition is induced by the tensor product of algebras.

There is a well-defined way to add and subtract homology classes, which makes into an abelian group, called the th homology group of.

This is a special case of the Chebotarev density theorem for the Nth cyclotomic field K. Indeed, the Galois group of K / Q is abelian and can be canonically identified with the group of invertible residue classes mod N. The splitting invariant of a prime p not dividing N is simply its residue class because the number of distinct primes into which p splits is φ ( N )/ m, where m is multiplicative order of p modulo N ; hence by the Chebotarev density theorem, primes are asymptotically uniformly distributed among different residue classes coprime to N.

The statement of the more general Chebotarev theorem is in terms of the Frobenius element of a prime ( ideal ), which is in fact an associated conjugacy class C of elements of the Galois group G. If we fix C then the theorem says that asymptotically a proportion | C |/| G | of primes have associated Frobenius element as C. When G is abelian the classes of course each have size 1.

Quite generally, the isomorphism classes of invertible sheaves on X themselves form an abelian group under tensor product.

is the abelian group of equivalence classes of non-degenerate ε-quadratic forms over R, where the underlying R-modules F are finitely generated free.

We simply take Ext ( A, B ) to be the set of equivalence classes of extensions of A by B, forming an abelian group under the Baer sum.

Thus homotopy classes from one spectrum to another form an abelian group.

The theorem states: Any linearly ordered abelian group can be embedded as an ordered subgroup of the additive group ℝ < sup > Ω </ sup > endowed with a lexicographical order, where ℝ is the additive group of real numbers ( with its standard order ), and Ω is the set of Archimedean equivalence classes of.

In particular, the category of finitely generated modules over a noetherian commutative ring is abelian ; in this way, abelian categories show up in commutative algebra.

In this way, abelian categories show up in algebraic topology and algebraic geometry.

For finitely generated abelian groups, rank is a strong invariant and every such group is determined up to isomorphism by its rank and torsion subgroup.

As we have seen, up to isomorphism, there are four groups, two abelian, and two non-abelian.

Some theorems on abelian varieties require the idea of abelian variety up to isogeny for their convenient statement.

To call this a direct sum decomposition, we should work in the category of abelian varieties up to isogeny.

The set of extensions up to equivalence is an abelian group that is a realization of the functor Ext ( A, B )

* For n = 3 the obvious analogue of the ( n − 1 )- dimensional representation is reducible – the permutation representation coincides with the regular representation, and thus breaks up into the three one-dimensional representations, as is abelian ; see the discrete Fourier transform for representation theory of cyclic groups.

The existence of these derived functors is supplied by homological algebra of the abelian category of sheaves ( and indeed this was a main reason to set up that theory ).

The effect of working on varieties with singular points is to show up a difference between Weil divisors ( in the free abelian group generated by codimension-one subvarieties ), and Cartier divisors coming from sections of invertible sheaves.

So the elements break up into equivalence classes, such that each equivalence class is the set of non-identity elements of a maximal abelian subgroup.

Complex multiplication of abelian varieties was an area opened up by the work of Shimura and Taniyama.