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A generalization of the concept of divisibility of integers without remainder (cf. [[Division|Division]]).
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A generalization of the concept of divisibility of integers without remainder (cf. [[Division]]).
  
An element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336501.png" /> of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336502.png" /> is divisible by another element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336503.png" /> if there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336504.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336505.png" />. One also says that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336506.png" /> divides <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336507.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336508.png" /> is said to be a multiple of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d0336509.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365010.png" /> is the divisor of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365011.png" />. The divisibility of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365012.png" /> by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365013.png" /> is denoted by the symbol <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365014.png" />.
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An element $a$ of a ring $A$ is divisible by another element $b \in A$ if there exists $c \in A$ such that $a = bc$. One also says that $b$ divides $a$ and $a$ is said to be a multiple of $b$, while $b$ is a divisor of $a$. The divisibility of $a$ by $b$ is denoted by the symbol $b | a$.
  
Any associative-commutative ring displays the following divisibility properties:
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Any [[Associative rings and algebras|associative]]-[[commutative ring]] displays the following divisibility properties:
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$$
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b | a \ \text{and}\ c | b \Rightarrow c | a \ ;
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$$
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$$
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b | a \Rightarrow cb | ca \ ;
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$$
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$$
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c | a \ \text{and}\ c |b \Rightarrow c | a \pm b \ .
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$$
  
<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/d/d033/d033650/d03365015.png" /></td> </tr></table>
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The last two properties are equivalent to saying that the set of elements divisible by $b$ forms an ideal, $bA$, of the ring $A$ (the principal ideal generated by the element $b$), which contains $b$ if $A$ is a ring with a unit element.
  
The last two properties are equivalent to saying that the set of elements divisible by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365016.png" /> forms an ideal, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365017.png" />, of the ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365018.png" /> (the principal ideal generated by the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365019.png" />), which contains <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365020.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365021.png" /> is a ring with a unit element.
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In an integral domain, elements $a$ and $b$ are simultaneously divisible by each other ($a|b$ and $b|a$) if and only if they are associated, i.e. $a \ ub$, where $u$ is an invertible element. Two associated elements generate the same principal ideal. The [[unit divisor]]s coincide, by definition, with invertible elements. A prime element in a ring is a non-zero element without proper divisors except unit divisors. In the ring of integers such elements are called primes (or prime numbers), and in a ring of polynomials they are known as irreducible polynomials. Rings in which — like in rings of integers or polynomials — there is unique decomposition into prime factors (up to unit divisors and the order of the sequence) are called factorial rings. For any finite set of elements in such a ring there exists a greatest common divisor and a lowest common multiple, both these quantities being uniquely determined up to unit divisors.
 
 
In an integral domain, elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365022.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365023.png" /> are simultaneously divisible by each other (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365024.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365025.png" />) if and only if they are associated, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365026.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/d/d033/d033650/d03365027.png" /> is an invertible element. Two associated elements generate the same principal ideal. The [[unit divisor]]s coincide, by definition, with invertible elements. A prime element in a ring is a non-zero element without proper divisors except unit divisors. In the ring of integers such elements are called primes (or prime numbers), and in a ring of polynomials they are known as irreducible polynomials. Rings in which — like in rings of integers or polynomials — there is unique decomposition into prime factors (up to unit divisors and the order of the sequence) are called factorial rings. For any finite set of elements in such a ring there exists a greatest common divisor and a lowest common multiple, both these quantities being uniquely determined up to unit divisors.
 
  
 
====References====
 
====References====
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<TR><TD valign="top">[3]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR>
 
<TR><TD valign="top">[3]</TD> <TD valign="top">  Z.I. Borevich,  I.R. Shafarevich,  "Number theory" , Acad. Press  (1966)  (Translated from Russian)  (German translation: Birkhäuser, 1966)</TD></TR>
 
</table>
 
</table>
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Revision as of 18:38, 25 September 2017

A generalization of the concept of divisibility of integers without remainder (cf. Division).

An element $a$ of a ring $A$ is divisible by another element $b \in A$ if there exists $c \in A$ such that $a = bc$. One also says that $b$ divides $a$ and $a$ is said to be a multiple of $b$, while $b$ is a divisor of $a$. The divisibility of $a$ by $b$ is denoted by the symbol $b | a$.

Any associative-commutative ring displays the following divisibility properties: $$ b | a \ \text{and}\ c | b \Rightarrow c | a \ ; $$ $$ b | a \Rightarrow cb | ca \ ; $$ $$ c | a \ \text{and}\ c |b \Rightarrow c | a \pm b \ . $$

The last two properties are equivalent to saying that the set of elements divisible by $b$ forms an ideal, $bA$, of the ring $A$ (the principal ideal generated by the element $b$), which contains $b$ if $A$ is a ring with a unit element.

In an integral domain, elements $a$ and $b$ are simultaneously divisible by each other ($a|b$ and $b|a$) if and only if they are associated, i.e. $a \ ub$, where $u$ is an invertible element. Two associated elements generate the same principal ideal. The unit divisors coincide, by definition, with invertible elements. A prime element in a ring is a non-zero element without proper divisors except unit divisors. In the ring of integers such elements are called primes (or prime numbers), and in a ring of polynomials they are known as irreducible polynomials. Rings in which — like in rings of integers or polynomials — there is unique decomposition into prime factors (up to unit divisors and the order of the sequence) are called factorial rings. For any finite set of elements in such a ring there exists a greatest common divisor and a lowest common multiple, both these quantities being uniquely determined up to unit divisors.

References

[1] E. Kummer, "Zur Theorie der komplexen Zahlen" J. Reine Angew. Math. , 35 (1847) pp. 319–326
[2] I.M. Vinogradov, "Elements of number theory" , Dover, reprint (1954) (Translated from Russian)
[3] Z.I. Borevich, I.R. Shafarevich, "Number theory" , Acad. Press (1966) (Translated from Russian) (German translation: Birkhäuser, 1966)
How to Cite This Entry:
Divisibility in rings. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Divisibility_in_rings&oldid=41963
This article was adapted from an original article by O.A. IvanovaS.A. Stepanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article