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. 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 and then and 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. Definition 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. Example: Groups For example, a group is an algebraic object consisting of a set together with a single binary operation, satisfying certain axioms. If is a group with operation , a congruence relation on is an equivalence relation on the elements of satisfying and for all . 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 and whenever and .