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  Encyclopedia of Keywords > Principal Ideal Domain > Euclidean Domain   Michael Charnine

Keywords and Sections
CONVERSE
INTEGRAL DOMAIN
PID
EUCLIDEAN ALGORITHM
PRINCIPAL IDEAL DOMAIN
EUCLIDEAN DOMAIN
Review of Short Phrases and Links

    This Review contains major "Euclidean Domain"- related terms, short phrases and links grouped together in the form of Encyclopedia article.

Definitions

  1. A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. (Web site)
  2. A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal. (Web site)
  3. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true. (Web site)
  4. Also, any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. (Web site)

Converse

  1. Any Euclidean domain is a unique factorization domain (UFD), although the converse is not true. (Web site)

Integral Domain

  1. A Euclidean domain is an integral domain which can be endowed with at least one Euclidean function. (Web site)

Pid

  1. Again, the converse is not true: not every PID is a Euclidean domain. (Web site)
  2. Also, any Euclidean domain is a PID; the algorithm used to calculate greatest common divisors may be used to find a generator of any ideal. (Web site)
  3. However, not every PID is a Euclidean domain; the ring furnishes a counterexample.

Euclidean Algorithm

  1. In abstract algebra, a Euclidean domain (also called a Euclidean ring) is a type of ring in which the Euclidean algorithm can be used. (Web site)
  2. If R is a Euclidean domain then a form of the Euclidean algorithm can be used to compute greatest common divisors.

Principal Ideal Domain

  1. Summary: Euclidean domain = principal ideal domain = unique factorization domain = integral domain = Commutative ring. (Web site)
  2. Despite this counterexample, the polynomial ring over any field is always a principal ideal domain and in fact, a Euclidean domain.

Euclidean Domain

  1. A Euclidean domain is always a principal ideal domain (PID), an integral domain in which every ideal is a principal ideal. (Web site)
  2. This is called "division with remainder" or " polynomial long division " and shows that the ring F[ X] is a Euclidean domain. (Web site)
  3. Euclidean domain A Euclidean domain is an integral domain in which a degree function is defined so that "division with remainder" can be carried out.

Categories

  1. Principal Ideal Domain
  2. Euclidean Algorithm
  3. Pid
  4. Integral Domain
  5. Counterexample
  6. Books about "Euclidean Domain" in Amazon.com

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  Short phrases about "Euclidean Domain"
  Originally created: April 04, 2011.
  Links checked: January 23, 2013.
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