In
algebra
Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics.
Elementary ...
, the kernel of a
homomorphism
In algebra, a homomorphism is a morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
(function that preserves the
structure
A structure is an arrangement and organization of interrelated elements in a material object or system, or the object or system so organized. Material structures include man-made objects such as buildings and machines and natural objects such a ...
) is generally the
inverse image of 0 (except for
groups
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
whose operation is denoted multiplicatively, where the kernel is the inverse image of 1). An important special case is the
kernel of a linear map. The
kernel of a matrix, also called the ''null space'', is the kernel of the linear map defined by the matrix.
The kernel of a homomorphism is reduced to 0 (or 1) if and only if the homomorphism is
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
, 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.
[See and .]
For some types of structure, such as
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s and
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, the possible kernels are exactly the substructures of the same type. This is not always the case, and, sometimes, the possible kernels have received a special name, such as
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
for groups and
two-sided ideals for
rings.
Kernels allow defining
quotient objects (also called
quotient algebras in
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...
, and
cokernels in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
). For many types of algebraic structure, the
fundamental theorem on homomorphisms (or
first isomorphism theorem) states that
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of a homomorphism is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
to the quotient by the kernel.
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
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
.
This article is a survey for some important types of kernels in algebraic structures.
Survey of examples
Linear maps
Let ''V'' and ''W'' be
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s over a
field (or more generally,
modules
Broadly speaking, modularity is the degree to which a system's components may be separated and recombined, often with the benefit of flexibility and variety in use. The concept of modularity is used primarily to reduce complexity by breaking a s ...
over a
ring) and let ''T'' be a
linear map
In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that ...
from ''V'' to ''W''. If 0
''W'' is the
zero vector of ''W'', then the kernel of ''T'' is the
preimage of the
zero subspace ; that is, the
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element 0
''W''. The kernel is usually denoted as , or some variation thereof:
:
Since a linear map preserves zero vectors, the zero vector 0
''V'' of ''V'' must belong to the kernel. The transformation ''T'' is injective if and only if its kernel is reduced to the zero subspace.
The kernel ker ''T'' is always a
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
of ''V''. Thus, it makes sense to speak of the
quotient space ''V''/(ker ''T''). The first isomorphism theorem for vector spaces states that this quotient space is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to the
image
An image is a visual representation of something. It can be two-dimensional, three-dimensional, or somehow otherwise feed into the visual system to convey information. An image can be an artifact, such as a photograph or other two-dimensio ...
of ''T'' (which is a subspace of ''W''). As a consequence, the
dimension
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coord ...
of ''V'' equals the dimension of the kernel plus the dimension of the image.
If ''V'' and ''W'' are
finite-dimensional
In mathematics, the dimension of a vector space ''V'' is the cardinality (i.e., the number of vectors) of a basis of ''V'' over its base field. p. 44, §2.36 It is sometimes called Hamel dimension (after Georg Hamel) or algebraic dimension to d ...
and
bases have been chosen, then ''T'' can be described by a
matrix
Matrix most commonly refers to:
* ''The Matrix'' (franchise), an American media franchise
** '' The Matrix'', a 1999 science-fiction action film
** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchi ...
''M'', and the kernel can be computed by solving the homogeneous
system of linear equations . In this case, the kernel of ''T'' may be identified to the
kernel of the matrix ''M'', also called "null space" of ''M''. The dimension of the null space, called the nullity of ''M'', is given by the number of columns of ''M'' minus the
rank of ''M'', as a consequence of the
rank–nullity theorem.
Solving
homogeneous differential equations often amounts to computing the kernel of certain
differential operator
In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...
s.
For instance, in order to find all twice-
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
s ''f'' from the
real line
In elementary mathematics, a number line is a picture of a graduated straight line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real number to a po ...
to itself such that
:
let ''V'' be the space of all twice differentiable functions, let ''W'' be the space of all functions, and define a linear operator ''T'' from ''V'' to ''W'' by
:
for ''f'' in ''V'' and ''x'' an arbitrary
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
.
Then all solutions to the differential equation are in ker ''T''.
One can define kernels for homomorphisms between modules over a
ring in an analogous manner. This includes kernels for homomorphisms between
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s as a special case. This example captures the essence of kernels in general
abelian categories
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of abe ...
; see
Kernel (category theory).
Group homomorphisms
Let ''G'' and ''H'' be
groups and let ''f'' be a
group homomorphism from ''G'' to ''H''. If ''e''
''H'' is the
identity element of ''H'', then the ''kernel'' of ''f'' is the preimage of the singleton set ; that is, the subset of ''G'' consisting of all those elements of ''G'' that are mapped by ''f'' to the element ''e''
''H''.
The kernel is usually denoted (or a variation). In symbols:
:
Since a group homomorphism preserves identity elements, the identity element ''e''
''G'' of ''G'' must belong to the kernel.
The homomorphism ''f'' is injective if and only if its kernel is only the singleton set . If ''f'' were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist
such that
and
. Thus
. ''f'' is a group homomorphism, so inverses and group operations are preserved, giving
; in other words,
, and ker ''f'' would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element
, then
, thus ''f'' would not be injective.
is a
subgroup of ''G'' and further it is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
. Thus, there is a corresponding
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the
first isomorphism theorem for groups.
In the special case of
abelian group
In mathematics, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on the order in which they are written. That is, the group operation is comm ...
s, there is no deviation from the previous section.
Example
Let ''G'' be the
cyclic group on 6 elements with
modular addition, ''H'' be the cyclic on 2 elements with modular addition, and ''f'' the homomorphism that maps each element ''g'' in ''G'' to the element ''g'' modulo 2 in ''H''. Then , since all these elements are mapped to 0
''H''. The quotient group has two elements: and . It is indeed isomorphic to ''H''.
Ring homomorphisms
Let ''R'' and ''S'' be
rings (assumed
unital) and let ''f'' be a
ring homomorphism from ''R'' to ''S''.
If 0
''S'' is the
zero element of ''S'', then the ''kernel'' of ''f'' is its kernel as linear map over the integers, or, equivalently, as additive groups. It is the preimage of the
zero ideal , which is, the subset of ''R'' consisting of all those elements of ''R'' that are mapped by ''f'' to the element 0
''S''.
The kernel is usually denoted (or a variation).
In symbols:
:
Since a ring homomorphism preserves zero elements, the zero element 0
''R'' of ''R'' must belong to the kernel.
The homomorphism ''f'' is injective if and only if its kernel is only the singleton set .
This is always the case if ''R'' is a
field, and ''S'' is not the
zero ring.
Since ker ''f'' contains the multiplicative identity only when ''S'' is the zero ring, it turns out that the kernel is generally not a
subring
In mathematics, a subring of ''R'' is a subset of a ring that is itself a ring when binary operations of addition and multiplication on ''R'' are restricted to the subset, and which shares the same multiplicative identity as ''R''. For those ...
of ''R.'' The kernel is a sub
rng, and, more precisely, a two-sided
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
of ''R''.
Thus, it makes sense to speak of the
quotient ring ''R''/(ker ''f'').
The first isomorphism theorem for rings states that this quotient ring is naturally isomorphic to the image of ''f'' (which is a subring of ''S''). (Note that rings need not be unital for the kernel definition).
To some extent, this can be thought of as a special case of the situation for modules, since these are all
bimodules over a ring ''R'':
* ''R'' itself;
* any two-sided ideal of ''R'' (such as ker ''f'');
* any quotient ring of ''R'' (such as ''R''/(ker ''f'')); and
* the
codomain of any ring homomorphism whose domain is ''R'' (such as ''S'', the codomain of ''f'').
However, the isomorphism theorem gives a stronger result, because ring isomorphisms preserve multiplication while module isomorphisms (even between rings) in general do not.
This example captures the essence of kernels in general
Mal'cev algebras.
Monoid homomorphisms
Let ''M'' and ''N'' be
monoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoid ...
s and let ''f'' be a
monoid homomorphism from ''M'' to ''N''. Then the ''kernel'' of ''f'' is the subset of the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
consisting of all those
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of elements of ''M'' whose components are both mapped by ''f'' to the same element in ''N''. The kernel is usually denoted (or a variation thereof). In symbols:
:
Since ''f'' is a
function, the elements of the form must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the
diagonal set .
It turns out that is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on ''M'', and in fact a
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
. Thus, it makes sense to speak of the
quotient monoid . The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of ''f'' (which is a
submonoid
In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0.
Monoids ...
of ''N''; for the congruence relation).
This is very different in flavour from the above examples. In particular, the preimage of the identity element of ''N'' is ''not'' enough to determine the kernel of ''f''.
Universal algebra
All the above cases may be unified and generalized in
universal algebra
Universal algebra (sometimes called general algebra) is the field of mathematics that studies algebraic structures themselves, not examples ("models") of algebraic structures.
For instance, rather than take particular groups as the object of study ...
.
General case
Let ''A'' and ''B'' be
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set o ...
s of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''.
Then the ''kernel'' of ''f'' is the subset of the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one t ...
''A'' × ''A'' consisting of all those
ordered pair
In mathematics, an ordered pair (''a'', ''b'') is a pair of objects. The order in which the objects appear in the pair is significant: the ordered pair (''a'', ''b'') is different from the ordered pair (''b'', ''a'') unless ''a'' = ''b''. (In con ...
s of elements of ''A'' whose components are both mapped by ''f'' to the same element in ''B''.
The kernel is usually denoted (or a variation).
In symbols:
:
Since ''f'' is a
function, the elements of the form (''a'', ''a'') must belong to the kernel.
The homomorphism ''f'' is injective if and only if its kernel is exactly the diagonal set .
It is easy to see that ker ''f'' is an
equivalence relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relatio ...
on ''A'', and in fact a
congruence relation
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done ...
.
Thus, it makes sense to speak of the
quotient algebra ''A''/(ker ''f'').
The
first isomorphism theorem in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a
subalgebra of ''B'').
Note that the definition of kernel here (as in the monoid example) doesn't depend on the algebraic structure; it is a purely
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
-theoretic concept.
For more on this general concept, outside of abstract algebra, see
kernel of a function.
Malcev algebras
In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special
neutral element (the
zero vector in the case of
vector space
In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
s, the
identity element in the case of
commutative groups, and the
zero element in the case of
rings or modules). The characteristic feature of a Malcev algebra is that we can recover the entire equivalence relation ker ''f'' from the
equivalence class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a ...
of the neutral element.
To be specific, let ''A'' and ''B'' be Malcev algebraic structures of a given type and let ''f'' be a homomorphism of that type from ''A'' to ''B''. If ''e''
''B'' is the neutral element of ''B'', then the ''kernel'' of ''f'' is the
preimage of the
singleton set ; that is, the
subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset of ...
of ''A'' consisting of all those elements of ''A'' that are mapped by ''f'' to the element ''e''
''B''.
The kernel is usually denoted (or a variation). In symbols:
:
Since a Malcev algebra homomorphism preserves neutral elements, the identity element ''e''
''A'' of ''A'' must belong to the kernel. The homomorphism ''f'' is injective if and only if its kernel is only the singleton set .
The notion of
ideal
Ideal may refer to:
Philosophy
* Ideal (ethics), values that one actively pursues as goals
* Platonic ideal, a philosophical idea of trueness of form, associated with Plato
Mathematics
* Ideal (ring theory), special subsets of a ring considered ...
generalises to any Malcev algebra (as
linear subspace
In mathematics, and more specifically in linear algebra, a linear subspace, also known as a vector subspaceThe term ''linear subspace'' is sometimes used for referring to flats and affine subspaces. In the case of vector spaces over the reals, l ...
in the case of vector spaces,
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
in the case of groups, two-sided ideals in the case of rings, and
submodule in the case of
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
s).
It turns out that ker ''f'' is not a
subalgebra of ''A'', but it is an ideal.
Then it makes sense to speak of the
quotient algebra ''G''/(ker ''f'').
The first isomorphism theorem for Malcev algebras states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a subalgebra of ''B'').
The connection between this and the congruence relation for more general types of algebras is as follows.
First, the kernel-as-an-ideal is the equivalence class of the neutral element ''e''
''A'' under the kernel-as-a-congruence. For the converse direction, we need the notion of
quotient in the Mal'cev algebra (which is
division on either side for groups and
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
for vector spaces, modules, and rings).
Using this, elements ''a'' and ''b'' of ''A'' are equivalent under the kernel-as-a-congruence if and only if their quotient ''a''/''b'' is an element of the kernel-as-an-ideal.
Algebras with nonalgebraic structure
Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations.
For example, one may consider
topological group
In mathematics, topological groups are logically the combination of groups and topological spaces, i.e. they are groups and topological spaces at the same time, such that the continuity condition for the group operations connects these two st ...
s or
topological vector spaces, which are equipped with a
topology
In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
.
In this case, we would expect the homomorphism ''f'' to preserve this additional structure; in the topological examples, we would want ''f'' to be a
continuous map.
The process may run into a snag with the quotient algebras, which may not be well-behaved.
In the topological examples, we can avoid problems by requiring that topological algebraic structures be
Hausdorff (as is usually done); then the kernel (however it is constructed) will be a
closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points. In a complete metric space, a ...
and the
quotient space will work fine (and also be Hausdorff).
Kernels in category theory
The notion of ''kernel'' in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
is a generalisation of the kernels of abelian algebras; see
Kernel (category theory).
The categorical generalisation of the kernel as a congruence relation is the ''
kernel pair
In category theory, a regular category is a category with finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture many properties of abelia ...
''.
(There is also the notion of
difference kernel, or binary
equaliser.)
See also
*
Kernel (linear algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the linear subspace of the domain of the map which is mapped to the zero vector. That is, given a linear map between two vector spaces and , the kernel o ...
*
Zero set
Notes
References
*
*
{{DEFAULTSORT:Kernel (Algebra)
Algebra
Isomorphism theorems
Linear algebra