quotient ring (original) (raw)

Definition. Let R be a ring and let I be a two-sided ideal (http://planetmath.org/Ideal) of R. To define the quotient ring R/I, let us first define an equivalence relation in R. We say that the elements a,b∈Rare equivalent, written as a∼b, if and only if a-b∈I. If a is an element of R, we denote the corresponding equivalence class by [a]. Thus [a]=[b] if and only if a-b∈I. The quotient ring of R modulo I is the setR/I={[a]|a∈R}, with a ring structure defined as follows. If [a],[b] are equivalence classes in R/I, then

• •
  [a]+[b]=[a+b],
• •
  [a]⋅[b]=[a⋅b].

Here a and b are some elements in R that represent [a] and [b]. By construction, every element in R/I has such a representative in R. Moreover, since I is closed under addition and multiplication, one can verify that the ring structure in R/I is well defined.

A common notation is a+I=[a] which is consistent with the notion of classes [a]=a⁢H∈G/H for a group G and a normal subgroup H.

Properties

Examples