His notion of abelian category is now the basic object of study in homological algebra.

The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups.

This is a functor which is contravariant in the first and covariant in the second argument, i. e. it is a functor Ab < sup > op </ sup > × Ab → Ab ( where Ab denotes the category of abelian groups with group homomorphisms ).

For an abelian category C, the inverse limit functor

If I is ordered ( not simply partially ordered ) and countable, and C is the category Ab of abelian groups, the Mittag-Leffler condition is a condition on the transition morphisms f < sub > ij </ sub > that ensures the exactness of.

More generally, if C is an arbitrary abelian category that has enough injectives, then so does C < sup > I </ sup >, and the right derived functors of the inverse limit functor can thus be defined.

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 ).

The adjective " abelian ", derived from his name, has become so commonplace in mathematical writing that it is conventionally spelled with a lower-case initial " a " ( e. g., abelian group, abelian category, and abelian variety ).

Consider the category Ab of abelian groups and group homomorphisms.

In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties.

The motivating prototype example of an abelian category is the category of abelian groups, Ab.

A category is abelian if

* A category is preadditive if it is enriched over the monoidal category Ab of abelian groups.

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:

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.

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

( For groups of low 2-rank the proof of this breaks down, because theorems such as the signalizer functor theorem only work for groups with elementary abelian subgroups of rank at least 3.

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.

Namely φ is universal for homomorphisms from G to an abelian group H: for any abelian group H and homomorphism of groups f: G → H there exists a unique homomorphism F: G < sup > ab </ sup > → H such that.

( pronounced " lim one ") such that if ( A < sub > i </ sub >, f < sub > ij </ sub >), ( B < sub > i </ sub >, g < sub > ij </ sub >), and ( C < sub > i </ sub >, h < sub > ij </ sub >) are three projective systems of abelian groups, and

If the ranges of the morphisms of the inverse system of abelian groups ( A < sub > i </ sub >, f < sub > ij </ sub >) are stationary, that is, for every k there exists j ≥ k such that for all i ≥ j: one says that the system satisfies the Mittag-Leffler condition.

In fact, it can be shown that any abelian category is equivalent to a full subcategory of such a category of modules ( Mitchell's embedding theorem ).

In abstract algebra, an abelian group ( G ,+) is called finitely generated if there exist finitely many elements x < sub > 1 </ sub >,..., x < sub > s </ sub > in G such that every x in G can be written in the form

Then every hom-set has an addition, endowing it with the structure of an abelian monoid, and such that the composition of morphisms is bilinear.

* An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal.

* An abelian category is a pre-abelian category such that every monomorphism and epimorphism is normal.

A group is called solvable if it has a subnormal series whose factor groups are all abelian, that is, if there are subgroups such that is normal in, and is an abelian group, for.

Consider the following commutative diagram in any abelian category ( such as the category of abelian groups or the category of vector spaces over a given field ) or in the category of groups.

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:

In mathematics ( especially algebraic topology and abstract algebra ), homology ( in part from Greek ὁμόιος homos " identical ") is a certain general procedure to associate a sequence of abelian groups or modules with a given mathematical object such as a topological space or a group.

Chain complexes form a category: A morphism from the chain complex ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) to the chain complex ( e < sub > n </ sub >: B < sub > n </ sub > → B < sub > n-1 </ sub >) is a sequence of homomorphisms f < sub > n </ sub >: A < sub > n </ sub > → B < sub > n </ sub > such that for all n. The n-th homology H < sub > n </ sub > can be viewed as a covariant functor from the category of chain complexes to the category of abelian groups ( or modules ).

If ( d < sub > n </ sub >: A < sub > n </ sub > → A < sub > n-1 </ sub >) is a chain complex such that all but finitely many A < sub > n </ sub > are zero, and the others are finitely generated abelian groups ( or finite dimensional vector spaces ), then we can define the Euler characteristic

Therefore rings and Z-algebras are equivalent concepts, in the same way that abelian groups and Z-modules are equivalent.

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.

There are two main notational conventions for abelian groups – additive and multiplicative.

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.

A field is therefore an algebraic structure 〈 F, +, ·, −, < sup >− 1 </ sup >, 0, 1 〉; of type 〈 2, 2, 1, 1, 0, 0 〉, consisting of two abelian groups:

Homomorphism groups: To every pair A, B of abelian groups one can assign the abelian group Hom ( A, B ) consisting of all group homomorphisms from A to B.

Unlike the homology groups and higher homotopy groups associated to a topological space, the fundamental group need not be abelian.

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.