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quotient ring
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(Definition)
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Definition. Let  be a ring and let  be a two-sided ideal of  . To define the quotient ring  , let us first define an equivalence relation in  . We say that the elements  are equivalent, written as  , if and only if  . If  is an element of  , we denote the corresponding equivalence class by ![$ [a]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img11.png "$ [a]$") . Thus ![$ [a]=[b]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img12.png "$ [a]=[b]$") if and only if  . The quotient ring of  modulo  is the set
![$ R/I=\{[a]\, \vert\, a\in R\}$](http://images.planetmath.org:8080/cache/objects/470/l2h/img16.png "$ R/I=\{[a]\, \vert\, a\in R\}$") , with a ring structure defined as follows. If ![$ [a],[b]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img17.png "$ [a],[b]$") are equivalence classes in  , then
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![$ [a]+[b] = [a+b]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img19.png "$ [a]+[b] = [a+b]$") ,
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![$ [a]\cdot [b]=[a\cdot b]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img20.png "$ [a]\cdot [b]=[a\cdot b]$") .
Here  and  are some elements in  that represent ![$ [a]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img24.png "$ [a]$") and ![$ [b]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img25.png "$ [b]$") . By construction, every element in  has such a representative in  . Moreover, since
 is closed under addition and multiplication, one can verify that the ring structure in  is well defined.
A common notation is ![$ a+I=[a]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img30.png "$ a+I=[a]$") which is consistent with the notion of classes
![$ [a]=aH\in G/H$](http://images.planetmath.org:8080/cache/objects/470/l2h/img31.png "$ [a]=aH\in G/H$") for a group  and a normal subgroup  .
- If
 is commutative, then  is commutative.
- The mapping
 ,
![$ a\mapsto [a]$](http://images.planetmath.org:8080/cache/objects/470/l2h/img37.png "$ a\mapsto [a]$") is a homomorphism, and is called the natural homomorphism.
- For a ring
 , we have
![$ R/R=\{[0]\}$](http://images.planetmath.org:8080/cache/objects/470/l2h/img39.png "$ R/R=\{[0]\}$") and  .
- Let
 , and let
 be the set of even numbers. Then  contains only two classes; one for even numbers, and one for odd numbers. Actually this quotient ring is a field. It is the only field with two elements (up to isomorphy) and is also denoted by
 .
- One way to construct complex numbers is to consider the field
![$ \mathbb{R}[T]/(T^2+1)$](http://images.planetmath.org:8080/cache/objects/470/l2h/img45.png "$ \mathbb{R}[T]/(T^2+1)$") . This field can viewed as the set of all polynomials of degree  with normal addition and
 , which is like complex multiplication.
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"quotient ring" is owned by mathwizard. [ full author list (3) | owner history (2) ]
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(view preamble)
Cross-references: complex multiplication, normal, degree, polynomials, complex numbers, field, odd numbers, contains, even numbers, homomorphism, mapping, commutative, normal subgroup, group, classes, consistent, well defined, multiplication, addition, closed under, represent, structure, equivalence class, equivalent, equivalence relation, ring
There are 23 references to this entry.
This is version 13 of quotient ring, born on 2001-10-23, modified 2007-11-29.
Object id is 470, canonical name is QuotientRing.
Accessed 10442 times total.
Classification:
| AMS MSC: | 16-00 (Associative rings and algebras :: General reference works ) |
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Pending Errata and Addenda
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