Kernel (algebra)

In the mathematical field of algebra is the kernel or null space of a mapping the set of elements that are mapped to 0, or more generally the neutral element. A particular significance of this term, if it is a group homomorphism, a linear map or a ring homomorphism in the figure. In this case, the mapping is injective if the core consists only of the identity element. The core is of central importance in the homomorphism.

Definition

  • Is a group homomorphism, then the amount
  • Is a linear map of vector spaces (or a homomorphism of abelian groups, or more generally, a homomorphism of modules ), it means the amount
  • Is a ring homomorphism, then the set

Importance

The core of a group homomorphism always contains the identity element, the core of a linear map always contains the zero vector. If it contains only the neutral element or the zero vector, it is called the core trivial.

A linear map and a homomorphism is injective if the core consists only of the zero vector and the neutral element (ie is trivial ).

The core is of central importance in the homomorphism.

Example ( linear map of vector spaces )

We consider the linear map defined by

Is defined. The picture is exactly the vectors of the form

From the zero vector, and others are not. The core is thus the set of

Geometrically, the core is in this case a straight line ( the z axis ), and therefore has a dimension of 1 The dimension of the core is also referred to as a defect, and can be calculated explicitly by means of the set rank.

Generalizations

In a class of objects is a neutral core of a morphism of the differential core of the pair ( f, 0), that is characterized by the following universal property:

  • For the inclusion.
  • Is a morphism, so that, factorized t clearly on ker f

Abstract formulated means that the core results from the universal morphism from Einbettungsfunktor of f in the corresponding object.

In universal algebra is the kernel of a map f induced by the equivalence relation on A, ie the amount. If A and B are algebraic structures of the same type ( for example, A and B are associations) and f is a homomorphism from A to B, then the equivalence relation ker ( f) is also a congruence. Conversely, it also easy to see that each congruence core of a homomorphism.

Cokernel

The cokernel is the dual concept to the core.

Is a linear map of vector spaces over a field, then the cokernel of f is the quotient of W is in the image of f

According to the cokernel of homomorphisms of abelian groups or modules defined over a ring.

The cokernel of the projection satisfies the following universal property: Every homomorphism is for those factored uniquely through q and. It results in a category with zero objects from the universal morphism from f to the corresponding object in Einbettungsfunktor of.

This property is also the definition in any of the categories for the Coker zero objects. In abelian categories of cokernel by the ratio in the image is the same.

Function (mathematics) Fundamental theorem on homomorphisms Null vector Equaliser (mathematics) Quotient space (linear algebra) Abelian category
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