Definition, Betydelse, Synonymer & Anagram | Engelska ordet ABELIAN

Definition av ABELIAN

1. (matematik) abelsk, kommutativ (om en grupp)

Exempel på hur man kan använda ABELIAN i en mening

• In mathematics, 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 the order in which they are written.
• 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.
• Under this addition, and with multiplication being defined as function composition, the endomorphisms of an abelian group form a ring (the endomorphism ring).
• 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.
• In mathematics, the Klein four-group is an abelian group with four elements, in which each element is self-inverse (composing it with itself produces the identity) and in which composing any two of the three non-identity elements produces the third one.
• 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.
• For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type.
• Every cyclic group is an abelian group (meaning that its group operation is commutative), and every finitely generated abelian group is a direct product of cyclic groups.
• The fundamental theorem of finitely generated abelian groups can be stated two ways, generalizing the two forms of the fundamental theorem of finite abelian groups.
• The most familiar examples of this construction occur when considering vector spaces (modules over a field) and abelian groups (modules over the ring Z of integers).
• The trivial object is also a zero object in Ab, the category of abelian groups, Rng the category of pseudo-rings, R-Mod, the category of modules over a ring, and K-Vect, the category of vector spaces over a field.
• every hom-set Hom(A,B) in C has the structure of an abelian group, and composition of morphisms is bilinear, in the sense that composition of morphisms distributes over the group operation.
• A category C is preadditive if all its hom-sets are abelian groups and composition of morphisms is bilinear; in other words, C is enriched over the monoidal category of abelian groups.
• 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.
• C is preadditive, that is enriched over the monoidal category of abelian groups (equivalently, all hom-sets in C are abelian groups and composition of morphisms is bilinear);.
• Abelian category, in category theory, a preabelian category in which every monomorphism is a kernel and every epimorphism is a cokernel.
• The snake lemma is valid in every abelian category and is a crucial tool in homological algebra and its applications, for instance in algebraic topology.
• In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions.
• In mathematics, especially homological algebra and other applications of abelian category theory, the short five lemma is a special case of the five lemma.
• In mathematics, especially homological algebra and other applications of abelian category theory, the five lemma is an important and widely used lemma about commutative diagrams.

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