quotient ring
Definition.
Let be a ring and let be a two-sided ideal (http://planetmath.org/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 . Thus if and only if .
The quotient ring of modulo is the set
, with a ring structure
defined as follows.
If are equivalence classes in , then
-
•
,
-
•
.
Here and are some elements in that represent and .
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 which is consistent with the notion of classes for a group and a normal subgroup .
Properties
-
1.
If is commutative
, then is commutative.
-
2.
The mapping , is a homomorphism
, and is called the natural homomorphism
(http://planetmath.org/NaturalHomomorphism).
Examples
-
1.
For a ring , we have and .
-
2.
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 .
-
3.
One way to construct complex numbers
is to consider the field . This field can viewed as the set of all polynomials
of degree with normal addition and , 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 |