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.

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.

For an example of an infinite abelian p-group, see Prüfer group, and for an example of an infinite simple p-group, see Tarski monster group.

For all abelian groups X, Y and Z we have a group isomorphism

For example, in the category Div of divisible ( abelian ) groups and group homomorphisms between them there are monomorphisms that are not injective: consider, for example, the quotient map q: Q → Q / Z, where Q is the rationals under addition, Z the integers ( also considered a group under addition ), and Q / Z is the corresponding quotient group.

For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function.

For any abelian group and any prime number p the set A < sub > Tp </ sub > of elements of A that have order a power of p is a subgroup called the p-power torsion subgroup or, more loosely, the p-torsion subgroup:

For each prime number p, this provides a functor from the category of abelian groups to the category of p-power torsion groups that sends every group to its p-power torsion subgroup, and restricts every homomorphism to the p-torsion subgroups.

For the present purposes, it suffices to consider this as a tensor product of Z-modules ( or equivalently of abelian groups ).

For example, the Hilbert class field, which is the maximal unramified abelian extension of a number field, corresponds to a very special class of ideals.

For example, the abelianized absolute Galois group G of is ( naturally isomorphic to ) an infinite product of the group of units of the p-adic integers taken over all prime numbers p, and the corresponding maximal abelian extension of the rationals is the field generated by all roots of unity.

For most abelian varieties, the algebra Hdg *( X ) is generated in degree one, so the Hodge conjecture holds.

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.

For example, the integers are a generator of the category of abelian groups ( since every abelian group is a quotient of a free abelian group ).

For C, and hence by the Lefschetz principle for every algebraically closed field of characteristic zero, the torsion group of an abelian variety of dimension g is isomorphic to ( Q / Z )< sup > 2g </ sup >.

For n ≥ 0, let C < sup > n </ sup >( G, M ) be the group of all functions from G < sup > n </ sup > to M. This is an abelian group ; its elements are called the ( inhomogeneous ) n-cochains.

For G abelian this is given by the Pontryagin duality theory.

For a non-negative integer k, the kth Betti number b < sub > k </ sub >( X ) of the space X is defined as the rank of the abelian group H < sub > k </ sub >( X ), the kth homology group of X. Equivalently, one can define it as the vector space dimension of H < sub > k </ sub >( X ; Q ), since the homology group in this case is a vector space over Q.

For abelian varieties such as A < sub > p </ sub >, there is a definition of local zeta-function available.

* For abelian varieties and curves there is an elementary description of ℓ-adic cohomology.

For abelian varieties the first ℓ-adic cohomology group is the dual of the Tate module, and the higher cohomology groups are given by its exterior powers.

For example, if one defines categories in terms of sets, that is, as sets of objects and morphisms ( usually called a small category ), or even locally small categories, whose hom-objects are sets, then there is no category of all sets, and so it is difficult for a category-theoretic formulation to apply to all sets.

For a small category of votes a quorum of 6000 was required, principally grants of citizenship, and here small coloured stones were used, white for yes and black for no.

For some schools of Hinduism and Buddhism the received textual tradition is an epistemological category equal to perception and inference ( although this is not necessarily true for some other schools ).

For the second exam, called the Principles and Practices, Part 2, or the Professional Engineering exam, candidates may select a particular engineering discipline's content to be tested on ; there is currently not an option for BME with this, meaning that any biomedical engineers seeking a license must prepare to take this examination in another category ( which does not affect the actual license, since most jurisdictions do not recognize discipline specialties anyway ).

For Gilbert Ryle ( 1949 ), a category ( in particular a " category mistake ") is an important semantic concept, but one having only loose affinities to an ontological category.

For example, a ( strict ) 2-category is a category together with " morphisms between morphisms ", i. e., processes which allow us to transform one morphism into another.

For example, an endomorphism of a vector space V is a linear map ƒ: V → V, and an endomorphism of a group G is a group homomorphism ƒ: G → G. In general, we can talk about endomorphisms in any category.

Limit functor: For a fixed index category J, if every functor J → C has a limit ( for instance if C is complete ), then the limit functor C < sup > J </ sup >→ C assigns to each functor its limit.

For endomorphisms and automorphisms, the descriptions above coincide with the category theoretic definitions ; the first three descriptions do not.

For example, most of the products that are explicitly listed as prohibited in the beer and malt liquor category in the Seattle area are ice beers.

" For her work, she won the Best Actress Silver Ombú category award at the 2002 Mar del Plata Film Festival.

For onsite banquet hosting, entertainment was still provided, but foodservice establishments of this category did not have long term contracts with Beijing opera troupes, so that performers varied for time to time, and topnotch performers usually did not perform here or any other foodservice establishments ranking lower.

For catering, different foodservice establishments of this category were incapable of handling topnotch catering on their own, but must join forces with other foodservice establishments of same ranking ( or lower ) to do the job.

* For each category of illness, the authors outline the conventional medical treatment, provide references to medical studies, and then discuss the macrobiotic approach.

For Internet Explorer's security settings, under the miscellaneous category, meta refresh can be turned off by the user, thereby disabling its redirect ability.

* For the general treatment of the concept of a product, see product ( category theory ), which describes how to combine two objects of some kind to create an object, possibly of a different kind.

For Anarchy, State, and Utopia ( 1974 ) Nozick received a National Book Award in category Philosophy and Religion.

For example, rather than asserting that sentences are constructed by a rule that combines a noun phrase ( NP ) and a verb phrase ( VP ) ( e. g. the phrase structure rule S → NP VP ), in categorial grammar, such principles are embedded in the category of the head word itself.

For example, the Fitting lemma shows that the endomorphism ring of a finite length indecomposable module is a local ring, so that the strong Krull-Schmidt theorem holds and the category of finite length modules is a Krull-Schmidt category.

Most recent designs are For The People and Empire of the Sun, each of which won the Charles S. Roberts Award for best game in their category.