Kernel (algebra)

A homomorphism φ from the group G to the group G' is illustrated, with the groups represented by a green oval on the left and a yellow circle on the right respectively. The kernel of φ is the blue circle on the left, as φ sends it to the identity element 1 of G'.

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.

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.

History

The mathematician Pontryagin is credited with using the word "kernel" in 1931 to describe the elements of a group that were sent to the identity element in another group.

Definition

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 {e_H}; 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 ker f (or a variation). In symbols:

$\ker f=\{g\in G:f(g)=e_{H}\}.$

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 {e_G}. If f were not injective, then the non-injective elements can form a distinct element of its kernel: there would exist a, b ∈ G such that a ≠ b and f(a) = f(b). Thus f(a)f(b)⁻¹ = e_H. f is a group homomorphism, so inverses and group operations are preserved, giving f(ab⁻¹) = e_H; in other words, ab⁻¹ ∈ 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 ≠ e_G ∈ ker f, then f(g) = f(e_G) = e_H, thus f would not be injective.

ker f is a subgroup of G and further it is a normal subgroup. Thus, there is a corresponding quotient group G / (ker f). 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.

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 additive groups. It is the preimage of the zero ideal {0_S}, 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 ker f (or a variation). In symbols:

$\operatorname {ker} f=\{r\in R:f(r)=0_{S}\}.$

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 {0_R}. 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 of R. The kernel is a subrng, and, more precisely, a two-sided ideal 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).

Linear maps

Main article: Kernel (linear algebra)

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 (or null space) is the preimage of the zero subspace {0_W}; 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 ker T, or some variation thereof:

$\ker T=\{\mathbf {v} \in V:T(\mathbf {v} )=\mathbf {0} _{W}\}.$

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.

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).

Monoid homomorphisms

Let M and N be monoids and let f be a monoid homomorphism from M to N. Then the kernel of f is the subset of the direct product M × M consisting of all those ordered pairs of elements of M whose components are both mapped by f to the same element in N. The kernel is usually denoted ker f (or a variation thereof). In symbols:

$\operatorname {ker} f=\left\{\left(m,m'\right)\in M\times M:f(m)=f\left(m'\right)\right\}.$

Since f is a function, the elements of the form (m, m) must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the diagonal set {(m, m) : m in M}.

It turns out that ker f is an equivalence relation on M, and in fact a congruence relation. Thus, it makes sense to speak of the quotient monoid M / (ker f). The first isomorphism theorem for monoids states that this quotient monoid is naturally isomorphic to the image of f (which is a submonoid of N; for the congruence relation).

This is very different in flavor from the above examples. In particular, the preimage of the identity element of N is not enough to determine the kernel of f.

Survey of examples

Group homomorphisms

Let G be the cyclic group on 6 elements {0, 1, 2, 3, 4, 5} with modular addition, H be the cyclic on 2 elements {0, 1} with modular addition, and f the homomorphism that maps each element g in G to the element g modulo 2 in H. Then ker f = {0, 2, 4} , since all these elements are mapped to 0_H. The quotient group G / (ker f) has two elements: {0, 2, 4} and {1, 3, 5}. It is indeed isomorphic to H.

Given a isomorphism $\varphi :G\to H$, one has $\ker \varphi =1$. On the other hand, if this mapping is merely a homomorphism where H is the trivial group, then $\varphi (g)=1$ for all $g\in G$, so thus $\ker \varphi =G$.

Let $\varphi :\mathbb {R} ^{2}\to \mathbb {R}$ be the map defined as $\varphi ((x,y))=x$. Then this is a homomorphism with the kernel consisting precisely the points of the form $(0,y)$. This mapping is considered the "projection onto the x-axis." A similar phenomenon occurs with the mapping $f:(\mathbb {R} ^{\times })^{2}\to \mathbb {R} ^{\times }$ defined as $f(a,b)=b$, where the kernel is the points of the form $(a,1)$

For a non-abelian example, let $Q_{8}$ denote the Quaternion group, and $V_{4}$ the Klein 4-group. Define a mapping $\varphi :Q_{8}\to V_{4}$ to be:

$\varphi (\pm 1)=1$

$\varphi (\pm i)=a$

$\varphi (\pm j)=b$

$\varphi (\pm k)=c$

Then this mapping is a homomorphism where $\ker \varphi =\{\pm 1\}$.

Ring homomorphisms

Consider the mapping $\varphi :\mathbb {Z} \to \mathbb {Z} /2\mathbb {Z}$ where the later ring is the integers modulo 2 and the map sends each number to its parity; 0 for even numbers, and 1 for odd numbers. This mapping turns out to be a homomorphism, and since the additive identity of the later ring is 0, the kernel is precisely the even numbers.

Let $\varphi :\mathbb {Q} [x]\to \mathbb {Q}$ be defined as $\varphi (p(x))=p(0)$. This mapping , which happens to be a homomorphism, sends each polynomial to its constant term. It maps a polynomial to zero if and only if said polynomial's constant term is 0. If we instead work with polynomials with real coefficients, then we again recieve a homomorphism with its kernel being the polynomials with constant term 0.

Linear maps

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 Mv = 0. 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

$xf''(x)+3f'(x)=f(x),$

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

$(Tf)(x)=xf''(x)+3f'(x)-f(x)$

for f in V and x an arbitrary real number. Then all solutions to the differential equation are in ker T.

Universal algebra

All the above cases may be unified and generalized in universal algebra. Let A and B be algebraic structures 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 A × A consisting of all those ordered pairs of elements of A whose components are both mapped by f to the same element in B. The kernel is usually denoted ker f (or a variation). In symbols:

$\operatorname {ker} f=\left\{\left(a,b\right)\in A\times A:f(a)=f\left(b\right)\right\}{\mbox{.}}$

The homomorphism f is injective if and only if its kernel is exactly the diagonal set {(a, a) : a ∈ A}, which is always at least contained inside the kernel.

It is easy to see that ker f is an equivalence relation on A, and in fact a congruence relation. 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.

Algebras with nonalgebraic structure

Sometimes algebras are equipped with a nonalgebraic structure in addition to their algebraic operations. For example, one may consider topological groups or topological vector spaces, which are equipped with a topology. 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 and the quotient space will work fine (and also be Hausdorff).

Kernels in category theory

The notion of kernel in category theory is a generalization of the kernels of abelian algebras; see Kernel (category theory). The categorical generalization of the kernel as a congruence relation is the kernel pair. (There is also the notion of difference kernel, or binary equalizer.)

See also

• Kernel (linear algebra)
• Zero set

References

Lang, Serge (2002). Algebra. Graduate Texts in Mathematics. Springer. ISBN 0-387-95385-X.