quotient ring

Definition. Let R be a ring and let I be a two-sided idealMathworldPlanetmath (http://planetmath.org/Ideal) of R. To define the quotient ringMathworldPlanetmath R/I, let us first define an equivalence relationMathworldPlanetmath in R. We say that the elements a,bR are equivalentMathworldPlanetmathPlanetmathPlanetmath, written as ab, if and only if a-bI. If a is an element of R, we denote the corresponding equivalence classMathworldPlanetmath by [a]. Thus [a]=[b] if and only if a-bI. The quotient ring of R modulo I is the set R/I={[a]|aR}, with a ring structureMathworldPlanetmath defined as follows. If [a],[b] are equivalence classes in R/I, then

  • [a]+[b]=[a+b],

  • [a][b]=[ab].

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 underPlanetmathPlanetmath additionPlanetmathPlanetmath 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]=aHG/H for a group G and a normal subgroupMathworldPlanetmath H.


  1. 1.

    If R is commutativePlanetmathPlanetmathPlanetmath, then R/I is commutative.

  2. 2.

    The mapping RR/I, a[a] is a homomorphismPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, and is called the natural homomorphismMathworldPlanetmathPlanetmath (http://planetmath.org/NaturalHomomorphism).


  1. 1.

    For a ring R, we have R/R={[0]} and R/{0}=R.

  2. 2.

    Let R=, and let I=2 be the set of even numbersMathworldPlanetmath. Then R/I contains only two classes; one for even numbers, and one for odd numbersMathworldPlanetmath. Actually this quotient ring is a field. It is the only field with two elements (up to isomorphy) and is also denoted by 𝔽2.

  3. 3.

    One way to construct complex numbersMathworldPlanetmathPlanetmath is to consider the field [T]/(T2+1). This field can viewed as the set of all polynomialsMathworldPlanetmathPlanetmathPlanetmath of degree 1 with normal addition and (a+bT)(c+dT)=ac-bd+(ad+bc)T, which is like complex multiplicationMathworldPlanetmath.

Title quotient ring
Canonical name QuotientRing
Date of creation 2013-03-22 11:52:32
Last modified on 2013-03-22 11:52:32
Owner mathwizard (128)
Last modified by mathwizard (128)
Numerical id 18
Author mathwizard (128)
Entry type Definition
Classification msc 16-00
Classification msc 81R12
Classification msc 20C30
Classification msc 81R10
Classification msc 81R05
Classification msc 20C32
Synonym difference ring
Synonym factor ring
Synonym residue-class ring
Related topic NaturalHomomorphism
Related topic QuotientRingModuloPrimeIdeal