Kernel (algebra)
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In
algebra Algebra () is one of the areas of mathematics, 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 mathem ...
, the kernel of a
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "sa ...
(function that preserves the structure) is generally the inverse image of 0 (except for groups 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 contraposi ...
, 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 com ...
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 ...
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 Group (mathematics), groups as ...
, and cokernels in category theory). 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 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 ...
s over a field (or more generally, modules 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 pr ...
from ''V'' to ''W''. If 0''W'' is the zero vector of ''W'', then the kernel of ''T'' is the
preimage In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or throug ...
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 o ...
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: : \ker T = \ . 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 of ''V''. Thus, it makes sense to speak of the
quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...
''V''/(ker ''T''). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic 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 coor ...
of ''V'' equals the dimension of the kernel plus the dimension of the image. If ''V'' and ''W'' are finite-dimensional 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 The rank–nullity theorem is a theorem in linear algebra, which asserts that the dimension of the domain of a linear map is the sum of its rank (the dimension of its image) and its ''nullity'' (the dimension of its kernel). p. 70, §2.1, Th ...
. Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. 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 :x f''(x) + 3 f'(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''(x) + 3 f'(x) - f(x) for ''f'' in ''V'' and ''x'' an arbitrary
real number In mathematics, a real number is a number that can be used to measurement, measure a ''continuous'' one-dimensional quantity such as a distance, time, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small var ...
. 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 com ...
s as a special case. This example captures the essence of kernels in general abelian categories; 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 In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
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: : \ker f = \ . 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 \in G such that a \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\left(ab^\right) = e_H; in other words, ab^ \in \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 \neq e_G \in \ker f, then f(g) = f(e_G) = e_H, thus ''f'' would not be injective. is a
subgroup In group theory, a branch of mathematics, given a 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, ''H'' is a subgrou ...
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 com ...
s, there is no deviation from the previous section.


Example

Let ''G'' be the
cyclic group In group theory, a branch of abstract algebra in pure mathematics, a cyclic group or monogenous group is a group, denoted C''n'', that is generated by a single element. That is, it is a set of invertible elements with a single associative bi ...
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 In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition prese ...
from ''R'' to ''S''. If 0''S'' is the
zero element In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive iden ...
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: : \operatorname f = \\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 field, and ''S'' is not the
zero ring In ring theory, a branch of mathematics, the zero ring or trivial ring is the unique ring (up to isomorphism) consisting of one element. (Less commonly, the term "zero ring" is used to refer to any rng of square zero, i.e., a rng in which f ...
. 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 of ''R''. Thus, it makes sense to speak of the
quotient ring In ring theory, a branch of abstract algebra, a quotient ring, also known as factor ring, difference ring or residue class ring, is a construction quite similar to the quotient group in group theory and to the quotient space in linear algebra. I ...
''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 In mathematics, the codomain or set of destination of a function is the set into which all of the output of the function is constrained to fall. It is the set in the notation . The term range is sometimes ambiguously used to refer to either ...
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. Monoids ...
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 co ...
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: : \operatorname f = \left\. 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 a ...
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 Group (mathematics), groups as ...
.


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 ...
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 co ...
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: : \operatorname f = \left\\mbox 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-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 ...
s, the
identity element In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures s ...
in the case of commutative groups, and the
zero element In mathematics, a zero element is one of several generalizations of 0, the number zero to other algebraic structures. These alternate meanings may or may not reduce to the same thing, depending on the context. Additive identities An additive iden ...
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 ...
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 In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or throug ...
of the
singleton set In mathematics, a singleton, also known as a unit set or one-point set, is a set with exactly one element. For example, the set \ is a singleton whose single element is 0. Properties Within the framework of Zermelo–Fraenkel set theory, t ...
; 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 o ...
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: : \operatorname f = \\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 ideal generalises to any Malcev algebra (as linear subspace 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 modules). 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 arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
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 ...
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 that is al ...
s, 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 ho ...
. 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 such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in va ...
. 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 spac ...
and the
quotient space Quotient space may refer to a quotient set when the sets under consideration are considered as spaces. In particular: *Quotient space (topology), in case of topological spaces * Quotient space (linear algebra), in case of vector spaces *Quotient ...
will work fine (and also be Hausdorff).


Kernels in category theory

The notion of ''kernel'' in category theory 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 abelian ...
''. (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 ...
*
Zero set In mathematics, a zero (also sometimes called a root) of a real-, complex-, or generally vector-valued function f, is a member x of the domain of f such that f(x) ''vanishes'' at x; that is, the function f attains the value of 0 at x, or ...


Notes


References

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