Abelian in a sentence

The class of Abelian categories is closed under several categorical constructions, for example, the category of chain complexes of an Abelian category, or the category of functors from a small category to an Abelian category are Abelian as well.

Free abelian group further The free abelian group on a set S is defined via its universal property in the analogous way, with obvious modifications: Consider a pair (F, φ), where F is an abelian group and φ: S → F is a function.

Note that there are examples of full subcategories of an abelian category which are themselves abelian but where the inclusion functor is not exact, so they are not abelian subcategories (see below).

The category of all finitely generated abelian groups is also an abelian category, as is the category of all finite abelian groups.

Additive groups of rings The additive group of a ring is an abelian group, but not all abelian groups are additive groups of rings (with nontrivial multiplication).

By definition, a ring is a monoid object in the category of abelian groups ; thus, an associative algebra is obtained by replacing the category of abelian groups with the category of modules.

Classification The fundamental theorem of finite abelian groups states that every finite abelian group G can be expressed as the direct sum of cyclic subgroups of prime -power order.

Every subgroup and factor group of a finitely generated abelian group is again finitely generated abelian.

Examples Ordered abelian groups In mathematics the language of ordered abelian groups has one constant symbol 0, one unary function symbol −, one binary function symbol +, and one binary relation symbol ≤.

Homomorphisms of abelian groups If G and H are abelian (i.

In particular, the real numbers are an abelian group under addition, and the nonzero real numbers are an abelian group under multiplication.

In this article, all modules will be assumed to be right unital modules over a ring R. Examples -modules are the same as abelian groups, so a simple Z-module is an abelian group which has no non-zero proper subgroups.

More specifically, every Z-module is an abelian group with its operation of addition, and every abelian group is a module over the ring of integers Z in a unique way.

Non-finitely generated abelian groups Note that not every abelian group of finite rank is finitely generated; the rank 1 group is one counterexample, and the rank-0 group given by a direct sum of countably infinitely many copies of is another one.

Preadditive categories, rings and modules A preadditive category is a category where the morphism sets form abelian groups and the composition of morphisms is bilinear ; examples are categories of abelian groups or modules.

The construction, detailed in the article on the Grothendieck group, is "universal", in that it has the universal property of being unique, and homomorphic to any other embedding of an abelian monoid in an abelian group.

The fundamental theorem of abelian groups states that every finitely generated abelian group is the direct product of finitely many finite primary cyclic and infinite cyclic groups.

Then are ideals of R and : as a direct sum of abelian groups (because for abelian groups finite products are the same as direct sums).

The preadditive categories most commonly studied are in fact Abelian categories; for example, Ab is an Abelian category.

The theory of Abelian groups is decidable, but that of non-Abelian groups is not.