thumb|A [[group homomorphism from the group to the group is illustrated, with the groups represented by a blue oval on the left and a yellow circle on the right, respectively. The kernel of is the red circle on the left, as sends it to the identity element 1 of .]]

thumb|300x300px|An example for a kernel - the linear operator <math> L : (x,y) \longrightarrow (x, x)</math> transforms all points on the <math> (x=0, y)</math> line to the zero point <math> (0,0)</math>, thus they form the kernel for the linear operator

In algebra, the kernel of a homomorphism is the relation describing how elements in the domain of the homomorphism become related in the image. A homomorphism is a function that preserves the underlying algebraic structure in the domain to its image.

When the algebraic structures involved have an underlying group structure, the kernel is taken to be the preimage of the group's identity element in the image, that is, it consists of the elements of the domain mapping to the image's identity. For example, the map that sends every integer to its parity (that is, 0 if the number is even, 1 if the number is odd) would be a homomorphism to the integers modulo 2, and its respective kernel would be the even integers which all have 0 as its parity.

For some types of structure, such as abelian groups and vector spaces, the possible kernels are exactly the substructures of the same type. This is not always the case, and some kernels have received a special name, such as normal subgroups for groups and two-sided ideals for rings. The concept of a kernel has been extended to structures such that the inverse image of a single element is not sufficient for deciding whether a homomorphism is injective. In these cases, the kernel is a congruence relation.

  1. Associative: <math>(a \cdot b) \cdot c = a \cdot (b \cdot c)</math>
  2. Identity: There is an <math>e \in G </math> such that <math>e \cdot a = a \cdot e = a</math>
  3. Inverses: There is an <math>a' \in G</math> for each <math>a \in G</math> such that <math>a \cdot a' = a' \cdot a = e</math>

A group is also called abelian if it also satisfies <math>a \cdot b = b \cdot a</math>. (For simplicity, the operation symbol <math>\cdot</math> is omitted.) Letting <math>e_H</math> be the identity element of <math>H</math>, then the kernel of <math>f</math> is the preimage of the singleton set <math>\{e_H\}</math>; that is, the subset of <math>G</math> consisting of all those elements of <math>G</math> that are mapped by <math>f</math> to the element <math>e_H</math>.

The kernel is usually denoted <math>\ker{f}</math> (or a variation).

<math>\ker{f}</math> is a subgroup of <math>G</math> and further it is a normal subgroup. Thus, there is a corresponding quotient group <math>G/\ker{f}</math>. This is isomorphic to <math>f(G)</math>, the image of <math>G</math> under <math>f</math> (which is a subgroup of <math>H</math> also), by the first isomorphism theorem for groups.

  1. <math>R</math> with <math>+</math> is an abelian group with identity <math>0</math>.
  2. Multiplication <math>\cdot</math> is associative.
  3. Distributive: <math>a \cdot (b + c) = a \cdot b + a \cdot c</math> and <math>(a + b) \cdot c = a \cdot c + b \cdot c</math> for all <math>a,b,c \in R</math>
  4. Multiplication <math>\cdot</math> has an identity element <math>1</math>.(a_1/\ker{f}, \ldots a_n/\ker{f}) = Q^A(a_1, \ldots a_n)/\ker{f}</math>

The first isomorphism theorem in universal algebra states that this quotient algebra is naturally isomorphic to the image of <math>f</math> (which is a subalgebra of <math>B</math>).

Category theory

Kernels of morphisms

Kernels can be generalized in categories that have zero objects. A category must satisfy having:

  • Objects <math>A \in \bold{C}</math>
  • Morphisms <math>f: A \to B</math>
  • Composition; if <math>f: A \to B</math> and <math>g: B \to C</math>, then denote their composition as <math> g \circ f: A \to C</math>
  • Associativity: if <math>f: A \to B</math>, <math>g: B \to C</math>, and <math>h: C \to D</math>, then <math>h \circ (g \circ f) = (h \circ g) \circ f</math>
  • An identity morphism <math>id_A: A \to A</math> where composition with it results in the same morphism; for <math>f: A \to B</math>, <math>f = f \circ id_A = id_B \circ f</math>

A morphism <math>f: A \to B</math> is an isomorphism when there exists a morphism <math>g: B \to A</math> such that <math>g \circ f</math> and <math>f \circ g</math> are the identity morphisms. If the zero object of a category is labeled <math>0</math>, then the composition of the morphisms <math>0: A \to 0 \to B</math> is the <math>0</math>-morphism from <math>A</math> to <math>B</math>.

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The kernel is denoted as <math>\ker f \to B</math>. The kernel is the limit of the diagram <math>B \xrightarrow{f} C \xleftarrow{} 0</math>. By reversing the direction of the morphisms and compositions given in the definition of a kernel, this defines the notion of a cokernel, denoted as <math>\text{coker} f</math>. The image (category theory) of a morphism is defined as <math>\text{im} f = \ker (\text{coker} f)</math> when the respective kernel/cokernel exist.

For abelian groups, the equalizer of two homomorphisms is the same as the equalizer between the difference of these two homomorphisms and the zero homomorphism, so the only equalizers that are needed to be considered in the category of abelian groups are the ones between any homomorphism <math>h: A \to B</math> and the zero homomorphism <math>0: A \to B</math>. The object of such an equalizer is (up to isomorphism) <math>\ker h</math>, the kernel of the homomorphism <math>h</math>, and the associated morphism is the inclusion map.

Kernel pairs

The kernel pair of a morphism <math>f: X \to Y</math> is defined as the pullback on this morphism paired with itself. It can be visualized with the commutative diagram:

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Kernels of functors

Functors between categories can also have a kernel. A (covariant) functor from a category <math>\bold{C}</math> to <math>\bold{D}</math>, denoted <math>F: \bold{C} \to \bold{D}</math>, maps objects and morphisms from <math>\bold{C}</math> to <math>\bold{D}</math> such that the following hold:

  1. If <math>f: A \to B</math>, then <math>F(f): F(A) \to F(B)</math>
  2. <math>F(g \circ f) = F(g) \circ F(f)</math>
  3. <math>F(id_A) = id_{F(A)}</math>

A congruence on a category <math>\bold{C}</math> is an equivalence relation <math>\sim</math> on morphisms where <math>f \sim g</math> implies they share the same domain and codomain, and furthermore <math>bfa \sim bga</math> for any applicable morphisms <math>a</math> and <math>b</math>. A congruence gives rise to an associated congruence category <math>\bold{C}^\sim</math> with the same objects as <math>\bold{C}</math> but with morphisms consisting of <math>\langle f, g \rangle</math> where <math>f \sim g</math>, composition being defined componentwise, and the identity morphism being <math>\widetilde{id_A} = \langle id_A, id_A \rangle</math>. Then a quotient category <math>\bold{C}/\sim</math> can be formed, where the objects are again the same as <math>\bold{C}</math>, the morphisms are the equivalence classes <math>[f]</math> under the congruence, the identity morphism being its associated equivalence class <math>[id_A]</math>, and composition defined as <math>[g] \circ [f] = [g \circ f]</math>. There are two projection functors from the congruence category to the original category, labeled as <math>p_1, p_2</math>, and there is a quotient functor <math>\pi</math> from the category to its quotient category acting as the coequalizer of the two projection functors.

A functor <math>F: \bold{C} \to \bold{D}</math> gives a congruence <math>\sim_F</math> where <math>f \sim_F g</math> if and only if they share the same domain and codomain, and furthermore <math>F(f) = F(g)</math>. The kernel of <math>F</math> is then denoted as the associated congruence category <math>\ker F = \bold{C}^{\sim_F}</math>.