Wikipedia / Kernel (algebra)
Wikipedia / Kernel (algebra)
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. The kernel of a homomorphism of group-like structures will be a singleton set that only contains the identity if and only if the homomorphism is injective, that is if the inverse image of every element consists of a single element. This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.
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.
Kernels allow defining quotient objects (also called quotient algebras in universal algebra). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is isomorphic to the quotient by the kernel.
Definition
Group homomorphisms
A group is a set with a binary operation:
A group is also called abelian.
be groups. A group homomorphism. be the identity element, then the kernel.
(or a variation). In symbols:
must belong to the kernel..
is a subgroup and further it is a normal subgroup. Thus, there is a corresponding quotient group also), by the first isomorphism theorem for groups.
Ring homomorphisms
A ring with identity (or unity) satisfying:
.
is associative.
.
A ring is commutative if the multiplication is commutative, and such a ring is a field has a multiplicative inverse. be rings. A ring homomorphism:
The kernel is the kernel as additive groups. It is the preimage of the zero ideal (or a variation). In symbols:
is a field is not the zero ring.
is the zero ring, it turns out that the kernel is generally not a subring. The kernel is a subrng, and, more precisely, a two-sided ideal. Thus, it makes sense to speak of the quotient ring).
Linear maps
Given a field, a vector space) with scalar multiplication:
be vector spaces. A linear map:
is the zero vector (or null space) is the preimage of the zero subspace; that is, the subset, or some variation thereof, and is symbolically defined as:
is injective if and only if its kernel is reduced to the zero subspace.
is always a linear subspace. Thus, it makes sense to speak of the quotient space. The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image). As a consequence, the dimension equals the dimension of the kernel plus the dimension of the image.
Module homomorphisms
be a ring. A module is defined exactly like a vector space over a field, using the same axioms, except the field is replaced with a ring. In fact, a module over a field is exactly the same as a vector space over a field.-modules. A module homomorphism is defined to be:
Every kernel is a submodule of the domain module, which means they always contain 0, the additive identity of the module. Kernels of abelian groups can be considered a particular kind of module kernel when the underlying ring is the integers.
Examples
Group homomorphisms
be the cyclic group with modular addition.
Given a isomorphism. On the other hand, if this mapping is merely a homomorphism where H.
. This mapping is considered the "projection onto the x-axis."
denote the Quaternion group the Klein 4-group to be:
.
denote the circle group, consisting of all complex numbers with absolute value (or modulus, with the group operation being multiplication..
The symmetric group that takes each permutation to the parity of the number of transpositions whose product is that permutation. The alternating group is the kernel of this homomorphism, consisting of the even permutations. The alternating group is a non-abelian simple group.
The determinant invertible matrices of the real numbers and called the general linear group (consisting of all non-zero real numbers), and the kernel of the determinant is called the special linear group.
is an automorphism to its respective inner automorphism as described, and the kernel of this homomorphism is the center. More generally, for every normal subgroup, with the kernel being the centralizer.
Ring homomorphisms
where the later ring is the integers modulo 2 and the map sends each number to its parity; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.
. This mapping, which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero if and only if said polynomial's constant term is 0. Polynomials with real coefficients can receive a similar homomorphism, with its kernel being the polynomials with constant term 0.
Linear maps
(the same is true for every kernel of a linear map).
represents the derivative operator on real polynomials will consist of the polynomials with deterivative equal to 0, that is the constant functions.
.
Quotient algebras
The kernel of a homomorphism can be used to define a quotient algebra to be the set of fibers, where a fiber is the set of points of the domain mapping to a single point in the range. is called the quotient group (or factor group), to be read as "G modulo K" or "G mod K"., and that the remaining elements are simply "translates" of the kernel, so the quotient group is obtained by "dividing out" the kernel.
where:
these sets are called the left and right cosets respectively, and can be defined in general for any arbitrary subgroup., which is well-defined regardless of the choice of representatives of the fibers.
According to the first isomorphism theorem, and such map is also well-defined.
For rings, modules, and vector spaces, and is well-defined. (possibly a field, which will also be well-defined.
Kernel structures
The structure of kernels allows for the building of quotient algebras from structures satisfying the properties of kernels. Any subgroup of a group by the set of all cosets., however this operation is well defined if and only if is closed under conjugation. Furthermore, the operation being well defined is sufficient for the quotient to be a group. Subgroups satisfying this property are called normal subgroups., so the normal subgroups are precisely the subgroups which are kernels. The closure under conjugation, however, gives a criterion for when a subgroup is a kernel for some homomorphism.
For a ring of the ring, which will be normal due to the ring's additive group being abelian, yields:
is also closed under multiplication by arbitrary elements on the right. that is closed under multiplication by any element of the ring is called an ideal. Analogously to normal subgroups, the ideals of a ring are precisely the kernels of homomorphisms.
Exact sequence
Kernels are used to define exact sequences of homomorphisms for groups and modules where each adjacent pair of modules and homomorphisms is exact.
when the zero module is the range. are exact if and only if the labeled homomorphism are injective, surjective, and an isomorphism respectively.
A particular kind of exact sequence is a short exact sequence fixed. is the kernel of .
Universal algebra
Kernels can be generalized in universal algebra for homomorphisms between any two algebraic structures. An operation is the arity is an indexed set of operations is the language, which also maps each operation symbol to their fixed arity (called the rank function). Two algebraic structures are similar when they share the same language, including their rank function.
:
The kernel, is the subset of the direct product consisting of all ordered pairs. In symbols:
, which is always contained inside the kernel. is an equivalence relation, and in fact a congruence relation. It makes sense to speak of the quotient algebra, with the set consisting of the equivalence classes as:
The first isomorphism theorem (which is a subalgebra).
Category theory
Kernels of morphisms
Kernels can be generalized in categories that have zero objects. A category must satisfy having:
are the identity morphisms. A zero object is an object of a category in which there exists exactly one morphism going to every object and exactly one morphism from every object. Any two zero objects are isomorphic to each other..
. This is illustrated in the commutative diagram:
. The kernel is the limit. By reversing the direction of the morphisms and compositions given in the definition of a kernel, this defines the notion of a cokernel. The image (category theory) when the respective kernel/cokernel exist.
The notions of kernels/cokernels gives rise to the definition of an abelian category. A category is additive when it has a zero object, products for any two objects, and the morphisms between any two fixed objects form an abelian group with composition distributing over the addition on this group. Morphisms in an additive category may be called homomorphisms. An additive category is then called an abelian category when every homomorphism has a kernel and cokernel, every monomorphism is the kernel of its cokernel, and every epimorphism is the cokernel of its kernel.
Equalizer
Kernels of morphisms can be generalized by the notion of an equalizer. Any equalizer morphism must be monic.
, and the associated morphism is the inclusion map. This example illustrates that equalizers are a generalization of a kernel of a morphism, in particular, the kernel of a morphism is the equalizer between the morphism with the respective zero morphism.
Kernel pairs
is defined as the pullback on this morphism paired with itself. It can be visualized with the commutative diagram:
Kernels of functors
Functors such that the following hold:
A congruence from the category to its quotient category acting as the coequalizer of the two projection functors.
.
See also
Notes
References
Sources