This means that the kernel can be viewed as a measure of the degree to which the homomorphism fails to be injective.
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.
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.
This article is a survey for some important types of kernels in algebraic structures.
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 0W is the zero vector of W, then the kernel of T is the preimage of the zero subspace {0W}; that is, the subset of V consisting of all those elements of V that are mapped by T to the element 0W.
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.
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.
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).
This example captures the essence of kernels in general Mal'cev algebras.
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 particular, the preimage of the identity element of N is not enough to determine the kernel of f. All the above cases may be unified and generalized in universal algebra.
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.
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).
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 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).