quotient ring

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 $a,b\in R$ are equivalent, written as $a\sim b$, if and only if $a-b\in I$. If $a$ is an element of $R$, we denote the corresponding equivalence class by $[a]$. Thus $[a]=[b]$ if and only if $a-b\in I$. The quotient ring of $R$ modulo $I$ is the set $R/I=\{[a]|a\in R\}$, with a ring structure defined as follows. If $[a],[b]$ are equivalence classes in $R/I$, then

• $[a]+[b]=[a+b]$,

• $[a]\cdot [b]=[a\cdot b]$.

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 $[a]=aH\in G/H$ for a group $G$ and a normal subgroup $H$.

Properties

1. If $R$ is commutative, then $R/I$ is commutative.

2. The mapping $R\to R/I$, $a\mapsto [a]$ is a homomorphism, and is called the natural homomorphism (http://planetmath.org/NaturalHomomorphism).

Examples

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

2. Let $R=\mathbb{Z}$, and let $I=2\mathbb{Z}$ be the set of even numbers. Then $R/I$ contains only two classes; one for even numbers, and one for odd numbers. Actually this quotient ring is a field. It is the only field with two elements (up to isomorphy) and is also denoted by ${𝔽}_2$.

3. One way to construct complex numbers is to consider the field $ℝ[T]/(T^2+1)$. This field can viewed as the set of all polynomials of degree $1$ with normal addition and $(a+bT)(c+dT)=ac-bd+(ad+bc)T$, which is like complex multiplication.

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