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 in the sense that algebraic operations done with equivalent elements will yield equivalent elements. Every congruence relation has a corresponding quotient structure, whose elements are the equivalence classes (or congruence classes) for the relation.

The definition of a congruence depends on the type of algebraic structure under consideration. Particular definitions of congruence can be made for groups, rings, vector spaces, modules, semigroups, lattices, and so forth. The common theme is that a congruence is an equivalence relation on an algebraic object that is compatible with the algebraic structure, in the sense that the operations are well-defined on the equivalence classes.

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.

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

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 ${\displaystyle 10}$,

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 ${\displaystyle 10}$.