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 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. 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 prototypical example of a congruence relation is congruence modulo n {displaystyle n} on the set of integers. For a given positive integer n {displaystyle n} , two integers a {displaystyle a} and b {displaystyle b} are called congruent modulo n {displaystyle n} , written if a − b {displaystyle a-b} is divisible by n {displaystyle n} (or equivalently if a {displaystyle a} and b {displaystyle b} have the same remainder when divided by n {displaystyle n} ). For example, 37 {displaystyle 37} and 57 {displaystyle 57} are congruent modulo 10 {displaystyle 10} , since 37 − 57 = − 20 {displaystyle 37-57=-20} is a multiple of 10, or equivalently since both 37 {displaystyle 37} and 57 {displaystyle 57} have a remainder of 7 {displaystyle 7} when divided by 10 {displaystyle 10} . Congruence modulo n {displaystyle n} (for a fixed n {displaystyle n} ) is compatible with both addition and multiplication on the integers. That is,