Quotient

In arithmetic, the quotient of two integers a and b is the result of dividing a by b, often written as a/b. It satisfies a = b q + r with remainder r, 0 ≤ r < |b| when b ≠ 0. If r = 0, a is a multiple of b and q is exact.

More generally, a quotient is the set of equivalence classes under an equivalence relation on a set X. The quotient X/~ consists of the classes, and operations on X that respect ~ descend to X/~.

In algebra, the quotient group G/N is formed from a group G and a normal subgroup N; its elements are cosets aN, and the product is (aN)(bN) = (ab)N. The natural projection π: G → G/N has kernel N.

In ring theory, if I is an ideal of a ring R, the quotient ring R/I has elements r+I; addition and multiplication are inherited from R. The projection π: R → R/I is a surjective ring homomorphism with kernel I.

In topology, a quotient space X/~ is formed by identifying points under ~; the quotient map q: X → X/~ is continuous and the quotient topology on X/~ is the finest topology making q continuous.