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.
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.
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.
The kernel of a ring homomorphism f from R to S consists of all those elements x of R for which f(x) = 0:
The isomorphism theorem mentioned above for groups and vector spaces remains valid in the case of rings.
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.
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[?].