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Kernel (algebra)

In mathematics, especially abstract algebra, the kernel of a homomorphism measures the degree to which the homomorphism fails to be injective.

In many cases, the kernel of a homomorphism is a subset of the domain of the homomorphism (specifically, those elements which are mapped to the identity element in the codomain). In more general contexts, the kernel is instead interpreted as a congruence relation on the domain (specifically, the relation of being mapped to the same image by the homomorphism). In either situation, the kernel is trivial if and only if the homomorphism is injective; in the first situation "trivial" means consisting of the identity element only, while in the second it means that the relation is equality.

In this article, we survey various definitions of kernel used for important types of homomorphisms.

Table of contents

Kernel of a group homomorphism

The kernel of a group homomorphism f from G to H consists of all those elements of G which are sent by f to the identity element eH of H. In formulas:

ker f := {x in G : f(x) = eH}.
The kernel is a normal subgroup of G.

One of the isomorphism theorems states that the factor group G/(ker f) is isomorphic to the image of f, the isomorphism being induced by f itself. A slightly more general statement is the fundamental theorem on homomorphisms.

The group homomorphism f is injective iff the kernel of f consists of the identity element of G only.

Kernel of a linear operator

If A is a linear transformation from a vector space V to a vector space W, then the kernel of A is defined as

ker A := {x in V : Ax = 0}.
The kernel is a subspace of the vector space V, and again the quotient space[?] V/(ker A) is isomorphic to the image of A; in particular, we have for the dimensions:
dim ker A = dim V - dim im A.
The operator A is injective if and only if ker A = {0}.

If V and W are finite-dimensional and bases have been chosen, then A can be described by a matrix M, and the kernel can be computed by solving the homogenous system of linear equations Mx = 0. In this representation, the kernel corresponds to the nullspace of M. The dimension of the nullspace, and hence of the kernel, is given by the number of columns of M minus the rank of M, a number also known as the nullity of M.

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 such that

xf''(x) + 3f'(x) = f(x),
one has to consider the kernel of a linear operator A from V to W, where V is the vector space of all twice differentible functions, W is the vector space of all functions, and for f in V, we define Af in W by requiring that
(Af)(x) = xf''(x) + 3f'(x) - f(x)
for every x.

One can define kernels for homomorphisms between modules of a ring in an analogous manner, and this example captures the essence of kernels in general abelian categories.

Kernel of a ring homomorphism

The kernel of a ring homomorphism f from R to S consists of all those elements x of R for which f(x) = 0:

ker f := {x in R : f(x) = 0}.
Such a kernel is always an ideal of R.

The isomorphism theorem mentioned above for groups and vector spaces remains valid in the case of rings.

Kernels in universal algebra

All the above cases are unified and generalized in universal algebra as follows: Given algebras A and B of the same type and a homomorphism f from A to B, the kernel of f is the congruence relation ~ on A defined as follows: Given elements x and y of A, let x ~ y iff f(x) = f(y). Clearly, this congruence degenerates to equality if and only if f is injective.

In the case of groups, if f is a group homomorphism from G to H, the two notions of kernel are related as follows: Given a and b in G, by definition a ~ b iff f(a) = f(b), which holds iff f(b)-1f(a) is the identity element eH of H. Since f is a homomorphism, this is true iff f(b-1a) is eH. So to know whether a ~ b, it's enough to keep track of the preimage of the identity of H, and this preimage is exactly the subgroup that we earlier called the kernel of f.

Essentially the same thing happens with vector spaces and rings and all other ideal supporting algebras[?], but in more general algebraic structures, kernels cannot be thought of as subsets but must be thought of as congruences.

Kernels in category theory

The notion of kernel of a morphism in category theory is a different generalization of the kernels of group and vector space homomorphisms. See kernel (category theory). The notion of kernel pair[?] is a further generalisation of the kernel as a congruence relation. There is also the notion of difference kernel[?], or binary equalizer[?].

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