quotient ring
Definition. Let $R$ be a ring and let $I$ be a twosided 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 $ab\in I$. If $a$ is an element of $R$, we denote the corresponding equivalence class^{} by $[a]$. Thus $[a]=[b]$ if and only if $ab\in I$. 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

•
$[a]+[b]=[a+b]$,

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$[a]\cdot [b]=[a\cdot b]$.
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 ${\mathbb{F}}_{2}$.

3.
One way to construct complex numbers^{} is to consider the field $\mathbb{R}[T]/({T}^{2}+1)$. This field can viewed as the set of all polynomials^{} of degree $1$ with normal addition and $(a+bT)(c+dT)=acbd+(ad+bc)T$, which is like complex multiplication^{}.
Title  quotient ring 
Canonical name  QuotientRing 
Date of creation  20130322 11:52:32 
Last modified on  20130322 11:52:32 
Owner  mathwizard (128) 
Last modified by  mathwizard (128) 
Numerical id  18 
Author  mathwizard (128) 
Entry type  Definition 
Classification  msc 1600 
Classification  msc 81R12 
Classification  msc 20C30 
Classification  msc 81R10 
Classification  msc 81R05 
Classification  msc 20C32 
Synonym  difference ring 
Synonym  factor ring 
Synonym  residueclass ring 
Related topic  NaturalHomomorphism 
Related topic  QuotientRingModuloPrimeIdeal 