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Difference between revisions of "Euclidean ring"

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An [[Integral domain|integral domain]] with an identity such that to each non-zero element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363701.png" /> of it corresponds a non-negative integer <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363702.png" /> satisfying the following requirement: For any two elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363703.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363704.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363705.png" /> one can find elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363707.png" /> such that
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An [[Integral domain|integral domain]] with an identity such that to each non-zero element $a$ of it corresponds a non-negative integer $n(a)$ satisfying the following requirement: For any two elements $a$ and $b$ with $b\neq0$ one can find elements $q$ and $r$ such that
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363708.png" /></td> </tr></table>
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$$a=bq+r,$$
  
where either <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e0363709.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e03637010.png" />.
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where either $r=0$ or $n(r)<n(b)$.
  
Every Euclidean ring is a [[Principal ideal ring|principal ideal ring]] and hence a [[Factorial ring|factorial ring]]; however, there exist principal ideal rings that are not Euclidean. Euclidean rings include the ring of integers (the absolute value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e03637011.png" /> plays the part of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e03637012.png" />), and also the ring of polynomials in one variable over a field (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/e/e036/e036370/e03637013.png" /> is the degree of the polynomial). In any Euclidean ring the [[Euclidean algorithm|Euclidean algorithm]] can be used to find the greatest common divisor of two elements.
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Every Euclidean ring is a [[Principal ideal ring|principal ideal ring]] and hence a [[Factorial ring|factorial ring]]; however, there exist principal ideal rings that are not Euclidean. Euclidean rings include the ring of integers (the absolute value $|a|$ plays the part of $n(a)$), and also the ring of polynomials in one variable over a field ($n(a)$ is the degree of the polynomial). In any Euclidean ring the [[Euclidean algorithm|Euclidean algorithm]] can be used to find the greatest common divisor of two elements.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A.G. Kurosh,  "Lectures on general algebra" , Chelsea  (1963)  (Translated from Russian)</TD></TR></table>

Revision as of 14:50, 15 April 2014

An integral domain with an identity such that to each non-zero element $a$ of it corresponds a non-negative integer $n(a)$ satisfying the following requirement: For any two elements $a$ and $b$ with $b\neq0$ one can find elements $q$ and $r$ such that

$$a=bq+r,$$

where either $r=0$ or $n(r)<n(b)$.

Every Euclidean ring is a principal ideal ring and hence a factorial ring; however, there exist principal ideal rings that are not Euclidean. Euclidean rings include the ring of integers (the absolute value $|a|$ plays the part of $n(a)$), and also the ring of polynomials in one variable over a field ($n(a)$ is the degree of the polynomial). In any Euclidean ring the Euclidean algorithm can be used to find the greatest common divisor of two elements.

References

[1] A.G. Kurosh, "Lectures on general algebra" , Chelsea (1963) (Translated from Russian)
How to Cite This Entry:
Euclidean ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Euclidean_ring&oldid=18798
This article was adapted from an original article by O.A. Ivanova (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article