In

_{''W''} is the _{''W''}. The kernel is usually denoted as , or some variation thereof:
:$\backslash ker\; T\; =\; \backslash \; .$
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

_{''H''} is the _{''H''}.
The kernel is usually denoted (or a variation). In symbols:
: $\backslash ker\; f\; =\; \backslash \; .$
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 $a,\; b\; \backslash in\; G$ such that $a\; \backslash neq\; b$ and $f(a)\; =\; f(b)$. Thus $f(a)f(b)^\; =\; e\_H$. ''f'' is a group homomorphism, so inverses and group operations are preserved, giving $f\backslash left(ab^\backslash right)\; =\; e\_H$; in other words, $ab^\; \backslash in\; \backslash ker\; f$, and ker ''f'' would not be the singleton. Conversely, distinct elements of the kernel violate injectivity directly: if there would exist an element $g\; \backslash neq\; e\_G\; \backslash in\; \backslash ker\; f$, then $f(g)\; =\; f(e\_G)\; =\; e\_H$, thus ''f'' would not be injective.
is a

_{''H''}. The quotient group has two elements: and . It is indeed isomorphic to ''H''.

_{''S''} is the _{''S''}.
The kernel is usually denoted (or a variation).
In symbols:
: $\backslash operatorname\; f\; =\; \backslash \backslash mbox$
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

_{''B''} is the neutral element of ''B'', then the ''kernel'' of ''f'' is the _{''B''}.
The kernel is usually denoted (or a variation). In symbols:
: $\backslash operatorname\; f\; =\; \backslash \backslash mbox$
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 _{''A''} under the kernel-as-a-congruence. For the converse direction, we need the notion of

algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In its most ge ...

, the kernel of a homomorphism
In algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

(function that preserves the structure
A structure is an arrangement and organization of interrelated elements in a material object or system
A system is a group of Interaction, interacting or interrelated elements that act according to a set of rules to form a unified whole.
...

) is generally the inverse image
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

of 0 (except for groups
A group is a number of people 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 identi ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s and vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

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
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

for groups and two-sided ideals for rings
Ring most commonly refers either to a hollow circular shape or to a high-pitched sound. It thus may refer to:
*Ring (jewellery), a circular, decorative or symbolic ornament worn on fingers, toes, arm or neck
Ring may also refer to:
Sounds
* Ri ...

.
Kernels allow defining quotient objectIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

s (also called quotient algebras in universal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...

, and cokernel
The cokernel of a linear mapping of vector spaces is the quotient space of the codomain of by the image of . The dimension of the cokernel is called the ''corank'' of .
Cokernels are dual to the kernels of category theory, hence the nam ...

s in category theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

). For many types of algebraic structure, the fundamental theorem on homomorphisms
In abstract algebra, the fundamental theorem on homomorphisms, also known as the fundamental homomorphism theorem, relates the structure of two objects between which a homomorphism is given, and of the Kernel (algebra), kernel and image of the hom ...

(or first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

) states that image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of a homomorphism is isomorphic
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). 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 an homomorphism is injective. In these cases, the kernel is a congruence relation
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

.
This article is a survey for some important types of kernels in algebraic structures.
Survey of examples

Linear maps

Let ''V'' and ''W'' bevector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s over a field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

(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 syst ...

over a ring) and let ''T'' be a linear map
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

from ''V'' to ''W''. If 0zero vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

of ''W'', then the kernel of ''T'' is the preimage
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the zero subspace ; that is, the subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of ''V'' consisting of all those elements of ''V'' that are mapped by ''T'' to the element 0linear subspace
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

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 Category (mathematics), categories invo ...

to the image
An image (from la, imago) is an artifact that depicts visual perception
Visual perception is the ability to interpret the surrounding environment (biophysical), environment through photopic vision (daytime vision), color vision, sco ...

of ''T'' (which is a subspace of ''W''). As a consequence, the dimension
In physics
Physics is the that studies , its , its and behavior through , and the related entities of and . "Physical science is that department of knowledge which relates to the order of nature, or, in other words, to the regular s ...

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 (linear algebra), basis of ''V'' over its base Field (mathematics), field. p. 44, §2.36 It is sometimes called Hamel dimension (after ...

and bases have been chosen, then ''T'' can be described by a matrix
Matrix or MATRIX may refer to:
Science and mathematics
* Matrix (mathematics), a rectangular array of numbers, symbols, or expressions
* Matrix (logic), part of a formula in prenex normal form
* Matrix (biology), the material in between a eukaryoti ...

''M'', and the kernel can be computed by solving the homogeneous system of linear equations
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

. 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
Rank is the relative position, value, worth, complexity, power, importance, authority, level, etc. of a person or object within a ranking
A ranking is a relationship between a set of items such that, for any two items, the first is either "rank ...

of ''M'', as a consequence of the rank–nullity theorem
The rank–nullity theorem is a theorem in linear algebra
Linear algebra is the branch of mathematics concerning linear equations such as:
:a_1x_1+\cdots +a_nx_n=b,
linear maps such as:
:(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n,
and ...

.
Solving homogeneous differential equation A differential equation
In mathematics, a differential equation is an functional equation, equation that relates one or more function (mathematics), functions and their derivatives. In applications, the functions generally represent physical qua ...

s often amounts to computing the kernel of certain differential operator
300px, A harmonic function defined on an annulus. Harmonic functions are exactly those functions which lie in the kernel of the Laplace operator, an important differential operator.
In mathematics, a differential operator is an Operator (mathe ...

s.
For instance, in order to find all twice-differentiable function
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

s ''f'' from the real line
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

to itself such that
:$x\; f\text{'}\text{'}(x)\; +\; 3\; f\text{'}(x)\; =\; f(x),$
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
:$(Tf)(x)\; =\; x\; f\text{'}\text{'}(x)\; +\; 3\; f\text{'}(x)\; -\; f(x)$
for ''f'' in ''V'' and ''x'' an arbitrary real number
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

.
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s as a special case. This example captures the essence of kernels in general abelian categories
In mathematics, an abelian category is a Category (mathematics), category in which morphisms and Object (category theory), objects can be added and in which Kernel (category theory), kernels and cokernels exist and have desirable properties. The mot ...

; see Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernel (algebra), kernels from algebra. Intuitivel ...

.
Group homomorphisms

Let ''G'' and ''H'' begroup
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

s and let ''f'' be a group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

from ''G'' to ''H''. If ''e''identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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''subgroup
In group theory, a branch of mathematics, given a group (mathematics), group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely ...

of ''G'' and further it is a normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

. Thus, there is a corresponding quotient group
A quotient group or factor group is a mathematical group (mathematics), 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 "factore ...

. This is isomorphic to ''f''(''G''), the image of ''G'' under ''f'' (which is a subgroup of ''H'' also), by the first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

for groups.
In the special case of abelian group
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

s, there is no deviation from the previous section.
Example

Let ''G'' be thecyclic group
In group theory
The popular puzzle Rubik's cube invented in 1974 by Ernő Rubik has been used as an illustration of permutation group">Ernő_Rubik.html" ;"title="Rubik's cube invented in 1974 by Ernő Rubik">Rubik's cube invented in 1974 by Er ...

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 0Ring homomorphisms

Let ''R'' and ''S'' be rings (assumed unital) and let ''f'' be aring homomorphism
In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function s ...

from ''R'' to ''S''.
If 0zero element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

, which is, the subset of ''R'' consisting of all those elements of ''R'' that are mapped by ''f'' to the element 0field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grassl ...

, and ''S'' is not the zero ring
In ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studies ...

.
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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

of ''R.'' The kernel is a sub rng, and, more precisely, a two-sided ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...

of ''R''.
Thus, it makes sense to speak of the quotient ring
In ring theory
In algebra, ring theory is the study of ring (mathematics), rings—algebraic structures in which addition and multiplication are defined and have similar properties to those operations defined for the integers. Ring theory studie ...

''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 bimoduleIn abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ri ...

s 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
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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'' bemonoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

s and let ''f'' be a monoid homomorphism
In abstract algebra, a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (ma ...

from ''M'' to ''N''. Then the ''kernel'' of ''f'' is the subset of the direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

consisting of all those ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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:
: $\backslash operatorname\; f\; =\; \backslash left\backslash .$
Since ''f'' is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

on ''M'', and in fact a congruence relation
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

. Thus, it makes sense to speak of the quotient monoid
In mathematics, a semigroup is an algebraic structure consisting of a set together with an associative binary operation
In mathematics, a binary operation or dyadic operation is a calculation that combines two elements (called operands) to pr ...

. 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.
Monoids are semigroups with identity. Such algebraic structures occur in several branches of mathematics.
...

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 inuniversal algebraUniversal algebra (sometimes called general algebra) is the field of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geomet ...

.
General case

Let ''A'' and ''B'' bealgebraic structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

''A'' × ''A'' consisting of all those ordered pair
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

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:
: $\backslash operatorname\; f\; =\; \backslash left\backslash \backslash mbox$
Since ''f'' is a function
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

, 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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

on ''A'', and in fact a congruence relation
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics) ...

.
Thus, it makes sense to speak of the quotient algebra ''A''/(ker ''f'').
The first isomorphism theorem
In mathematics, specifically abstract algebra, the isomorphism theorems (also known as Noether's isomorphism theorems) are theorems that describe the relationship between quotients, homomorphisms, and subobjects. Versions of the theorems exist for ...

in general universal algebra states that this quotient algebra is naturally isomorphic to the image of ''f'' (which is a subalgebraIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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-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 specialneutral element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

(the zero vector
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

in the case of vector space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

s, the identity element
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

in the case of commutative group
In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are ...

s, and the zero element
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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''preimage
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of the singleton set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

; that is, the subset
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

of ''A'' consisting of all those elements of ''A'' that are mapped by ''f'' to the element ''e''ideal
Ideal may refer to:
Philosophy
* Ideal (ethics)
An ideal is a principle
A principle is a proposition or value that is a guide for behavior or evaluation. In law
Law is a system
A system is a group of Interaction, interacting ...

generalises to any Malcev algebra (as linear subspace
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and t ...

in the case of vector spaces, normal subgroup
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathematics), ...

in the case of groups, two-sided ideals in the case of rings, and submodule
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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
* Modula ...

s).
It turns out that ker ''f'' is not a subalgebraIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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''quotient
In arithmetic
Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne' ...

in the Mal'cev algebra (which is division
Division or divider may refer to:
Mathematics
*Division (mathematics), the inverse of multiplication
*Division algorithm, a method for computing the result of mathematical division
Military
*Division (military), a formation typically consisting ...

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 considertopological group
In mathematics, topological groups are logically the combination of Group (mathematics), groups and Topological space, topological spaces, i.e. they are groups and topological spaces at the same time, such that the Continuous function, continui ...

s or topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space (an Abstra ...

s, with are equipped with a topology
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

.
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
In mathematics, a continuous function is a function (mathematics), function such that a continuous variation (that is a change without jump) of the argument of a function, argument induces a continuous variation of the Value (mathematics), value o ...

.
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 (as is usually done); then the kernel (however it is constructed) will be a closed set
In geometry
Geometry (from the grc, γεωμετρία; ' "earth", ' "measurement") is, with , one of the oldest branches of . It is concerned with properties of space that are related with distance, shape, size, and relative position of ...

and the quotient space will work fine (and also be Hausdorff).
Kernels in category theory

The notion of ''kernel'' incategory theory
Category theory formalizes mathematical structure
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and ...

is a generalisation of the kernels of abelian algebras; see Kernel (category theory)
In category theory and its applications to other branches of mathematics, kernels are a generalization of the kernels of group homomorphisms, the kernels of module homomorphisms and certain other kernel (algebra), kernels from algebra. Intuitivel ...

.
The categorical generalisation of the kernel as a congruence relation is the ''kernel pair
In category theory, a regular category is a category with limit (category theory), finite limits and coequalizers of a pair of morphisms called kernel pairs, satisfying certain ''exactness'' conditions. In that way, regular categories recapture many ...

''.
(There is also the notion of difference kernel, or binary equalizer (mathematics), equaliser.)
See also

* Kernel (linear algebra) * Zero setNotes

References

* * {{DEFAULTSORT:Kernel (Algebra) Algebra Isomorphism theorems Linear algebra