Kernel (algebra)
In
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.[1]
For some types of structure, such as
Kernels allow defining
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
Since a linear map preserves zero vectors, the zero vector 0V 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
If V and W are
Solving homogeneous differential equations often amounts to computing the kernel of certain differential operators. For instance, in order to find all twice-
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
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
Group homomorphisms
Let G and H be groups and let f be a group homomorphism from G to H. If eH is the identity element of H, then the kernel of f is the preimage of the singleton set {eH}; that is, the subset of G consisting of all those elements of G that are mapped by f to the element eH.
The kernel is usually denoted ker f (or a variation). In symbols:
Since a group homomorphism preserves identity elements, the identity element eG of G must belong to the kernel.
The homomorphism f is injective if and only if its kernel is only the singleton set {eG}. 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 = eH. f is a group homomorphism, so inverses and group operations are preserved, giving f(ab−1) = eH; 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 ≠ eG ∈ ker f, then f(g) = f(eG) = eH, 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.
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 {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 0H. The quotient group G / (ker f) has two elements: {0, 2, 4} and {1, 3, 5}. It is indeed isomorphic to H.
Ring homomorphisms
Algebraic structure → Ring theory Ring theory |
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Let R and S be
Since a ring homomorphism preserves zero elements, the zero element 0R of R must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {0R}. 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).
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 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
Monoid homomorphisms
Let M and N be
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
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.
General case
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:
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 {(a, a) : a ∈ A}.
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
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
Malcev algebras
This section may be confusing or unclear to readers. In particular, this section cannot be understood, as referring to a structure which is different from Malcev algebra and is not defined nor linked. (December 2016) |
In the case of Malcev algebras, this construction can be simplified. Every Malcev algebra has a special
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 eB is the neutral element of B, then the kernel of f is the
Since a Malcev algebra homomorphism preserves neutral elements, the identity element eA of A must belong to the kernel. The homomorphism f is injective if and only if its kernel is only the singleton set {eA}.
The notion of
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 eA under the kernel-as-a-congruence. For the converse direction, we need the notion of quotient in the Mal'cev algebra (which is division on either side for groups and subtraction 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
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
See also
- Kernel (linear algebra)
- Zero set
Notes
- ^ See Dummit & Foote (2004) and Lang (2002).
References
- Dummit, David S.; Foote, Richard M. (2004). Abstract Algebra (3rd ed.). ISBN 0-471-43334-9.
- ISBN 0-387-95385-X.