Core (algebra)

This article deals with the concept of the core of a figure; for further uses of this term see core .

In algebra , the core of a mapping is used to indicate how much the mapping deviates from injectivity . The exact definition depends on which algebraic structures are being considered. For example, the core of a linear mapping between vector spaces and those vectors in that are mapped onto the zero vector in ; so it is the solution set of the homogeneous linear equation and is here also called null space . In this case, then exactly injective when the core only of the zero vector in there. Analogous definitions apply to group and ring homomorphisms . The core is of central importance in the homomorphism theorem . ${\displaystyle f\colon V\to W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle V}$ ${\displaystyle W}$ ${\displaystyle f(x)=0}$ ${\displaystyle f}$ ${\displaystyle V}$

definition

• If there is a group homomorphism , the set becomes ${\displaystyle f\colon G\rightarrow H}$

${\displaystyle \operatorname {Kern} f:=\{g\in G\mid f(g)=e_{H}\in H\}}$

of all elements of that are mapped onto the neutral element of , called the core of . He is a normal divisor in . ${\displaystyle G}$ ${\displaystyle e_{H}}$ ${\displaystyle H}$ ${\displaystyle f}$ ${\displaystyle G}$

• If a linear mapping of vector spaces (or more generally a module homomorphism ) is called the set ${\displaystyle f\colon V\to W}$

${\displaystyle \operatorname {Kern} f:=\{v\in V\mid f(v)=0\in W\}}$

the core of . It is a subspace (more generally a submodule ) of . ${\displaystyle f}$ ${\displaystyle V}$

• If a ring homomorphism is the set ${\displaystyle f\colon A\to B}$

${\displaystyle \operatorname {Kern} f:=\{a\in A\mid f(a)=0\}}$

the core of . He is a two-sided ideal in . ${\displaystyle f}$ ${\displaystyle A}$

In English, will be held and or (for English. Kernel written). ${\displaystyle \operatorname {Kern} }$ ${\displaystyle \ker }$ ${\displaystyle \operatorname {Ker} }$

meaning

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

A linear mapping or a homomorphism is injective if and only if the kernel consists only of the zero vector or the neutral element (i.e. is trivial).

The core is of central importance in the homomorphism theorem .

Example (linear mapping of vector spaces)

We consider the linear map that goes through ${\displaystyle f\colon \mathbb {R} ^{3}\to \mathbb {R} ^{3}}$

${\displaystyle f(x)={\begin{pmatrix}1&0&0\\0&1&0\\0&0&0\end{pmatrix}}{\begin{pmatrix}x_{1}\\x_{2}\\x_{3}\end{pmatrix}}={\begin{pmatrix}x_{1}\\x_{2}\\0\end{pmatrix}}}$

is defined. The figure forms exactly the vectors of the shape ${\displaystyle f}$

${\displaystyle x={\begin{pmatrix}0\\0\\\lambda \end{pmatrix}},\lambda \in \mathbb {R} }$

on the zero vector and others do not. So the core of is the amount ${\displaystyle f}$

${\displaystyle \operatorname {Kern} f=\left\{{\begin{pmatrix}0\\0\\\lambda \end{pmatrix}},\lambda \in \mathbb {R} \right\}}$.

In this case, the core is geometrically a straight line (the axis) and accordingly has dimension 1. The dimension of the core is also referred to as a defect and can be calculated explicitly with the aid of the ranking . ${\displaystyle z}$

Generalizations

Universal algebra

In universal algebra , the core of a mapping is the induced equivalence relation on , i.e. the set . If and are algebraic structures of the same type (for example and are lattices) and there is a homomorphism from to , then the equivalence relation is also a congruence relation . Conversely, it is easy to show that every congruence relation is the core of a homomorphism. The mapping is injective if and only if the identity relation is on . ${\displaystyle f\colon A\to B}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle \operatorname {Kern} (f):=\{(x,y)\in A\times A\mid f(x)=f(y)\}}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle f}$ ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle \operatorname {Kern} (f)}$ ${\displaystyle f}$ ${\displaystyle \operatorname {Kern} (f)}$ ${\displaystyle \{(a,a)\mid a\in A\}}$ ${\displaystyle A}$

Category theory

In a category with zero objects , a core of a morphism is the difference core of the pair , that is, characterized by the following universal property : ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f\colon X\to Y}$ ${\displaystyle (f,0)}$

• The following applies to inclusion . ${\displaystyle i\colon \operatorname {Kern} f\to X}$ ${\displaystyle fi=0}$
• Is a morphism such that is so clearly factored over . ${\displaystyle t\colon T\to X}$ ${\displaystyle ft=0}$ ${\displaystyle t}$ ${\displaystyle \operatorname {Kern} f}$

In more abstract terms, this means that the core results from the universal morphism of the embedding function from in to the corresponding object. ${\displaystyle ({\mathcal {C}}\downarrow 0)}$ ${\displaystyle ({\mathcal {C}}\downarrow {\mathcal {C}})}$ ${\displaystyle f}$

Coke core

The Kokern , alternative spelling Cokern , is the dual term for the core.

Is a linear map of vector spaces over a body , then the Coker of the quotient of after the image of . ${\displaystyle f\colon V\to W}$ ${\displaystyle f}$ ${\displaystyle W}$ ${\displaystyle f}$

Correspondingly, the coke core for homomorphisms of Abelian groups or modules is defined over a ring.

The coke core with the projection fulfills the following universal property : Every homomorphism for which applies clearly factors over and it applies . In a category with zero objects, it results from the universal morphism of the corresponding object to the embedding functor of in . ${\displaystyle q\colon W\to \operatorname {coker} f}$ ${\displaystyle t\colon W\to T}$ ${\displaystyle tf=0}$ ${\displaystyle q}$ ${\displaystyle qf=0}$ ${\displaystyle {\mathcal {C}}}$ ${\displaystyle f}$ ${\displaystyle (0\downarrow {\mathcal {C}})}$ ${\displaystyle ({\mathcal {C}}\downarrow {\mathcal {C}})}$

This property is also the definition for the coke in any categories with zero objects . In Abelian categories the coke corresponds to the quotient according to the picture .

Web links

• Calculating the core of a matrix (example) ( Memento from March 4, 2016 in the Internet Archive )