quotiëntobject

A quotiëntobject, in the context of abstract algebra, is a mathematical object formed by taking an existing algebraic structure and dividing it by an equivalence relation. More specifically, if you have a set S and an equivalence relation ~ on S, the quotiëntobject is the set of all equivalence classes of S under ~. Each equivalence class is a subset of S containing all elements related to each other by ~. The set of these equivalence classes is denoted by S/~.

This concept is fundamental in understanding how to "factor out" certain properties of an algebraic structure. For example, in group theory, if G is a group and N is a normal subgroup of G, then the set of left cosets of N in G forms a group under a naturally defined operation. This group of cosets is the quotiëntgroup (or factor group) of G by N, and its elements are the equivalence classes formed by the relation g1 ~ g2 if and only if g1g2^-1 is in N.

Similarly, in ring theory, if R is a ring and I is an ideal of R, the set of cosets of I in R forms a ring, known as the quotiëntring (or factor ring) of R by I. The elements of this quotiëntring are the equivalence classes defined by a ~ b if and only if a - b is in I. The process of forming a quotiëntobject essentially creates a new structure where the "distinguishing feature" captured by the equivalence relation is essentially collapsed or "divided out."