Editing Congruence relation (section)

== Congruences of groups, and normal subgroups and ideals ==
In the particular case of [[group (mathematics)|groups]], congruence relations can be described in elementary terms as follows:
If ''G'' is a group (with [[identity element]] ''e'' and operation *) and ~ is a [[binary relation]] on ''G'', then ~ is a congruence whenever:
# [[Given any]] element ''a'' of ''G'', {{nowrap|''a'' ~ ''a''}} ('''[[reflexive relation|reflexivity]]''');
# Given any elements ''a'' and ''b'' of ''G'', [[material conditional|if]] {{nowrap|''a'' ~ ''b''}}, then {{nowrap|''b'' ~ ''a''}} ('''[[Symmetric relation|symmetry]]''');
# Given any elements ''a'', ''b'', and ''c'' of ''G'', if {{nowrap|''a'' ~ ''b''}} [[logical conjunction|and]] {{nowrap|''b'' ~ ''c''}}, then {{nowrap|''a'' ~ ''c''}} ('''[[Transitive relation|transitivity]]''');
# Given any elements ''a'', ''a''′, ''b'', and ''b''′ of ''G'', if {{nowrap|''a'' ~ ''a''′}} and {{nowrap|''b'' ~ ''b''′}}, then {{nowrap|''a'' * ''b'' ~ ''a''′ * ''b''′}};
# Given any elements ''a'' and ''a''′ of ''G'', if {{nowrap|''a'' ~ ''a''′}}, then {{nowrap|''a''<sup>−1</sup> ~ ''a''′<sup>−1</sup>}} (this is implied by the other four,<ref group=note>Since ''a''′<sup>−1</sup> = ''a''′<sup>−1</sup> * ''a'' * ''a''<sup>−1</sup> ~ ''a''′<sup>−1</sup> * ''a''′ * ''a''<sup>−1</sup> = ''a''<sup>−1</sup></ref> so is strictly redundant).

Conditions 1, 2, and 3 say that ~ is an [[equivalence relation]].

A congruence ~ is determined entirely by the set {{nowrap|{{mset|''a'' ∈ ''G'' | ''a'' ~ ''e''}}}} of those elements of ''G'' that are congruent to the identity element, and this set is a [[normal subgroup]].
Specifically, {{nowrap|''a'' ~ ''b''}} if and only if {{nowrap|''b''<sup>−1</sup> * ''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''.

=== Ideals of rings and the general case ===
A similar trick allows one to speak of kernels in [[ring (mathematics)|ring theory]] as [[ideal (ring theory)|ideals]] instead of congruence relations, and in [[module (mathematics)|module theory]] as [[submodule]]s instead of congruence relations.

A more general situation where this trick is possible is with [[Omega-group]]s (in the general sense allowing operators with multiple arity). But this cannot be done with, for example, [[monoid]]s, so the study of congruence relations plays a more central role in monoid theory.