Kernel (algebra)

In universal algebra, given algebras A and B of a certain type and a homomorphism f from A to B, the kernel of f is a certain congruence on A. Specifically, given elements a and b of A, let a ~ b iff f(a) = f(b). Then the relation ~ is the kernel of f.

Specifically, this is what a kernel amounts to in the case of groups, the most familiar kind of algebra: If G and H are groups and f is a group homomorphism from G to H, then the kernel ~ is defined as above. But there is another way to think of it. Given a and b in G, a ~ b iff f(a) = f(b), but that holds iff f(b)^-1 * f(a) is the identity element e_H of H. Since f is a homomorphism, this is true iff f(b^-1a) is e_H. So to know whether a ~ b, it's enough to keep track of the subgroup {x in G : f(x) = e} of G consisting of those elements of G that are mapped by f to the identity of H. a ~ b iff b^-1a belongs to that subgroup. It is this subgroup, not the relation ~, that is called kernel in group theory. But in more general universal algebra, kernels cannot be thought of as subalgebras but must be thought of as congruences.

A similar construction is possible with other kinds of algebras that have a group structure, possibly among other structures. Examples of these are rings, modules and vector spaces.

There is a more general notion of kernel of a morphism in category theory.