quotienten

Quotienten are mathematical objects derived by partitioning a set according to an equivalence relation or by performing division. In elementary arithmetic, the quotient of two numbers a and b is the unique number q such that a = bq when b ≠ 0. More generally, a quotient is the representative element of an equivalence class in which elements that are considered equivalent are identified as a single item.

In algebra, quotient constructions form new structures from existing ones by collapsing elements that are related. A quotient group G/N is obtained from a group G by identifying elements that differ by an element of a normal subgroup N. A quotient ring R/I is formed by collapsing elements that differ by an element of an ideal I. A quotient vector space V/W identifies vectors that differ by a member of a subspace W. The natural projection maps, such as π:G→G/N, are surjective and universal for maps that are constant on equivalence classes. The First Isomorphism Theorem links quotients to kernels of homomorphisms.

Quotients also appear in topology and analysis. A quotient space X/~ is formed by partitioning X into equivalence classes and equipping the set of classes with the final topology, such that a map from X to another space is continuous if and only if its composition with the projection is continuous. In number theory, quotients such as Z/nZ provide finite rings with arithmetic modulo n. The concept underpins many constructions in geometry, algebraic topology, and algebraic geometry.