General

The general notion of a congruence relation can be formally defined in the context of universal algebra, a field which studies ideas common to all algebraic structures. In this setting, a relation ${\displaystyle R}$ on a given algebraic structure is called compatible if

for each ${\displaystyle n}$ and each ${\displaystyle n}$-ary operation ${\displaystyle \mu }$ defined on the structure: whenever ${\displaystyle a_{1}\mathrel {R} a'_{1}}$ and ... and ${\displaystyle a_{n}\mathrel {R} a'_{n}}$, then ${\displaystyle \mu (a_{1},\ldots ,a_{n})\mathrel {R} \mu (a'_{1},\ldots ,a'_{n})}$.

A congruence relation on the structure is then defined as an equivalence relation that is also compatible.^[3][4]

Basic example

The prototypical example of a congruence relation is congruence modulo ${\displaystyle n}$ on the set of integers. For a given positive integer ${\displaystyle n}$, two integers ${\displaystyle a}$ and ${\displaystyle b}$ are called congruent modulo ${\displaystyle n}$, written

${\displaystyle a\equiv b{\pmod {n}}}$

if ${\displaystyle a-b}$ is divisible by ${\displaystyle n}$ (or equivalently if ${\displaystyle a}$ and ${\displaystyle b}$ have the same remainder when divided by ${\displaystyle n}$).

For example, ${\displaystyle 37}$ and ${\displaystyle 57}$ are congruent modulo ${\displaystyle 10}$,

${\displaystyle 37\equiv 57{\pmod {10}}}$

since ${\displaystyle 37-57=-20}$ is a multiple of 10, or equivalently since both ${\displaystyle 37}$ and ${\displaystyle 57}$ have a remainder of ${\displaystyle 7}$ when divided by ${\displaystyle 10}$.

Congruence modulo ${\displaystyle n}$ (for a fixed ${\displaystyle n}$) is compatible with both addition and multiplication on the integers. That is,

if

${\displaystyle a_{1}\equiv a_{2}{\pmod {n}}}$ and ${\displaystyle b_{1}\equiv b_{2}{\pmod {n}}}$

then

${\displaystyle a_{1}+b_{1}\equiv a_{2}+b_{2}{\pmod {n}}}$ and ${\displaystyle a_{1}b_{1}\equiv a_{2}b_{2}{\pmod {n}}}$

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

Example: Groups

For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If ${\displaystyle G}$ is a group with operation ${\displaystyle \ast }$, a congruence relation on ${\displaystyle G}$ is an equivalence relation ${\displaystyle \equiv }$ on the elements of ${\displaystyle G}$ satisfying

${\displaystyle g_{1}\equiv g_{2}\ \ \,}$ and ${\displaystyle \ \ \,h_{1}\equiv h_{2}\implies g_{1}\ast h_{1}\equiv g_{2}\ast h_{2}}$

for all ${\displaystyle g_{1},g_{2},h_{1},h_{2}\in G}$. For a congruence on a group, the equivalence class containing the identity element is always a normal subgroup, and the other equivalence classes are the other cosets of this subgroup. Together, these equivalence classes are the elements of a quotient group.

Example: Rings

When an algebraic structure includes more than one operation, congruence relations are required to be compatible with each operation. For example, a ring possesses both addition and multiplication, and a congruence relation on a ring must satisfy

${\displaystyle r_{1}+s_{1}\equiv r_{2}+s_{2}}$ and ${\displaystyle r_{1}s_{1}\equiv r_{2}s_{2}}$

whenever ${\displaystyle r_{1}\equiv r_{2}}$ and ${\displaystyle s_{1}\equiv s_{2}}$. For a congruence on a ring, the equivalence class containing 0 is always a two-sided ideal, and the two operations on the set of equivalence classes define the corresponding quotient ring.

Ideals of rings and the general case

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.

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