Integral domain: Difference between revisions
New page: An '''integral domain''' is a commutative domain. More explicitly a ring, <math>R</math>, is an integral domain if: * it is commutative, * <math>0\neq 1<... |
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* The [[p-adic numbers|p-adic integers]], <math>\mathbb{Z}_p</math>. | * The [[p-adic numbers|p-adic integers]], <math>\mathbb{Z}_p</math>. | ||
* For any integral domain, <math>R</math>, the [[polynomial ring]] <math>R[x]</math> is also an integral domain. | * For any integral domain, <math>R</math>, the [[polynomial ring]] <math>R[x]</math> is also an integral domain. | ||
* Any finite integral domain is a field. | |||
{{stub}} | {{stub}} | ||
[[Category:Ring theory]] | |||
Latest revision as of 16:43, 16 March 2012
An integral domain is a commutative domain.
More explicitly a ring, 
, is an integral domain if:
- it is commutative,
(where 
and 
are the additive and multiplicative identities, respectively)- and it contains no zero divisors (i.e. there are no nonzero
such that 
).
(where 
and 
are the additive and multiplicative identities, respectively)
such that 
).Examples
Some common examples of integral domains are:
- The ring
of integers. - Any field.
- The p-adic integers,
. - For any integral domain,
, the polynomial ring ![$R[x]$](//latex.artofproblemsolving.com/1/4/d/14de719c0fff2385124cfbcbd01a2dbb7b9b3300.png)
is also an integral domain. - Any finite integral domain is a field.
of 
.![$R[x]$](http://latex.artofproblemsolving.com/1/4/d/14de719c0fff2385124cfbcbd01a2dbb7b9b3300.png)
is also an integral domain.This article is a stub. Help us out by expanding it.
