Congruence relation

In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

Basic example

The prototypical example of a congruence relation is congruence modulo on the set of integers. For a given positive integer , two integers and are called congruent modulo , written

if is divisible by (or equivalently if and have the same remainder when divided by ).

for example, and are congruent modulo ,

since is a multiple of 10, or equivalently since both and have a remainder of when divided by .

Congruence modulo (for a fixed ) is compatible with both addition and multiplication on the integers. That is,

if

then

The corresponding addition and multiplication of equivalence classes is known as modular arithmetic. From the point of view of abstract algebra, congruence modulo is a congruence relation on the ring of integers, and arithmetic modulo occurs on the corresponding quotient ring.