Definition, Betydelse & Synonymer | Engelska ordet INJECTIVE

Definition av INJECTIVE

1. (matematik) injektiv

Exempel på hur man kan använda INJECTIVE i en mening

• Equivalently, a set is countable if there exists an injective function from it into the natural numbers; this means that each element in the set may be associated to a unique natural number, or that the elements of the set can be counted one at a time, although the counting may never finish due to an infinite number of elements.
• Many arguments involving finite sets rely on the pigeonhole principle, which states that there cannot exist an injective function from a larger finite set to a smaller finite set.
• The epimorphisms in Set are the surjective maps, the monomorphisms are the injective maps, and the isomorphisms are the bijective maps.
• The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935.
• However, in the more general context of category theory, the definition of a monomorphism differs from that of an injective homomorphism.
• Every map that is injective, continuous and either open or closed is an embedding; however there are also embeddings that are neither open nor closed.
• In the context of abstract algebra or universal algebra, a monomorphism is an injective homomorphism.
• If f is injective, this topology is canonically identified with the subspace topology of S, viewed as a subset of X.
• If an operator is not injective (so there is some nonzero x with T(x) = 0), then it is clearly not invertible.
• In category theory, a branch of mathematics, the concept of an injective cogenerator is drawn from examples such as Pontryagin duality.
• Every subgroup of a free abelian group is itself free abelian; this fact allows a general abelian group to be understood as a quotient of a free abelian group by "relations", or as a cokernel of an injective homomorphism between free abelian groups.
• The third property characterizes free and transitive actions, the onto character coming from transitivity, and then the injective character follows from the action being free.
• It suffices to show the map π is injective, since for *-morphisms of C*-algebras injective implies isometric.
• For example, if f : M → M is a surjective R-endomorphism of a finitely generated module M, then f is also injective, and hence is an automorphism of M.
• Usually the morphisms induced by inclusion in this theorem are not themselves injective, and the more precise version of the statement is in terms of pushouts of groups.
• Given a finitely generated module over a commutative ring, the pairing may be injective (hence "nondegenerate" in the above sense) but not unimodular.
• The monomorphisms in Ab are the injective group homomorphisms, the epimorphisms are the surjective group homomorphisms, and the isomorphisms are the bijective group homomorphisms.
• This is a special feature of hereditary rings like the integers Z: the direct sum of injective modules is injective because the ring is Noetherian, and the quotients of injectives are injective because the ring is hereditary, so any submodule generated by injective modules is injective.
• Over a Noetherian ring, every injective module is uniquely a direct sum of indecomposable modules, and their structure is well understood.
• If g is an essential monomorphism with domain X and an injective codomain G, then G is called an injective hull of X.

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