kernel (universal algebra)

In algebra, the kernel of a homomorphism (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, 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 groups and vector spaces, 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 for groups and two-sided ideals for rings.

Kernels allow defining quotient objects (also called quotient algebras in universal algebra, and cokernels in category theory). For many types of algebraic structure, the fundamental theorem on homomorphisms (or first isomorphism theorem) states that image of a homomorphism is isomorphic to the quotient by the kernel.

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.

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 spaces over a field (or more generally, modules over a ring) and let ''T'' be a linear map 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 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 ''V''/(ker ''T''). The first isomorphism theorem for vector spaces states that this quotient space is naturally isomorphic to the image of ''T'' (which is a subspace of ''W''). As a consequence, the dimension 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 ''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 operators.
For instance, in order to find all twice- differentiable functions ''f'' from the real line 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.
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 groups 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 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 of ''G'' and further it is a normal subgroup. Thus, there is a corresponding quotient group . 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 groups, 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'' cons