Difference between revisions of "Integral domain"

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m (Examples)
 
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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.
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* Any finite integral domain is a field.
  
 
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[[Category:Ring theory]]
 
[[Category:Ring theory]]

Latest revision as of 17:43, 16 March 2012

An integral domain is a commutative domain.

More explicitly a ring, $R$, is an integral domain if:

  • it is commutative,
  • $0\neq 1$ (where $0$ and $1$ are the additive and multiplicative identities, respectively)
  • and it contains no zero divisors (i.e. there are no nonzero $x,y\in R$ such that $xy = 0$).

Examples

Some common examples of integral domains are:

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