Difference between revisions of "Euclidean ring"
From Encyclopedia of Mathematics
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| − | An [[Integral domain|integral domain]] with an identity such that to each non-zero element | + | {{TEX|done}} |
| + | 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 | ||
| − | + | $$a=bq+r,$$ | |
| − | where either | + | 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 | + | 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
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