Table of contents |

2 In universal algebra 3 Group theory 4 Ring theory 5 General case of kernels |

The prototypical example is modular arithmetic:
If *a*_{1} = *a*_{2} (mod *n*) and *b*_{1} = *b*_{2} (mod *n*), then *a*_{1} + *b*_{1} = *a*_{2} + *b*_{2} (mod *n*) and *a*_{1}*b*_{1} = *a*_{2}*b*_{2} (mod *n*). This turns the equivalence (mod *n*) into a congruence on the ring of all integers.

The idea is generalized in universal algebra:
A congruence relation on an algebra *A* is a subset of the direct product *A* × *A* that is both an equivalence relation on *A* and a subalgebra of *A* × *A*.

Congruences typically arise as kernelss of homomorphisms, and in fact every congruence is the kernel of *some* homomorphism:
For a given congruence ~ on *A*, the set *A*/~ of equivalence classes can be given the structure of an algebra in a natural fashion, the quotient algebra.
Furthermore, the function that maps every element of *A* to its equivalence class is a homomorphism, and the kernel of this homomorphism is ~.

In the particular case of groups, congruence relations can be described in elementary terms as follows:
If *G* is a group (with identity element *e*) and ~ is a binary relation on *G*, then ~ is a congruence whenever:

- Given any element
*a*of*G*,*a*~*a*; - Given any elements
*a*and*b*of*G*, if*a*~*b*, then*b*~*a*; - Given any elements
*a*,*b*, and*c*of*G*, if*a*~*b*and*b*~*c*, then*a*~*c*; *e*~*e*;- Given any elements
*a*and*a*' of*G*, if*a*~*a*', then*a*^{−1}~*a*'^{−1}; - Given any elements
*a*,*a*',*b*, and*b*' of*G*, if*a*~*a*' and*b*~*b*', then*a***b*~*a*' **b*'.

Notice that such a congruence ~ is determined entirely by the set {*a* ∈ *G* : *a* ~ *e*} of those elements of *G* that are congruent to the identity element, and this set is a normal subgroup.
Specifically, *a* ~ *b* iff *b*^{−1} * *a* ~ *e*.
So instead of talking about congruences on groups, people usually speak in terms of normal subgroups of them; in fact, every congruence corresponds uniquely to some normal subgroup of *G*.
This is what makes it possible to speak of kernels in group theory as subgroups, while in more general universal algebra, kernels are congruences.

A similar trick allows one to speak of kernels in ring theory as ideals instead of congruence relations, and in module theory as submodules instead of congruence relations.