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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.
 
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.
  
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 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.
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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====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kummer,  "Zur Theorie der komplexen Zahlen"  ''J. Reine Angew. Math.'' , '''35'''  (1847)  pp. 319–326</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</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>
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<table>
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<TR><TD valign="top">[1]</TD> <TD valign="top">  E. Kummer,  "Zur Theorie der komplexen Zahlen"  ''J. Reine Angew. Math.'' , '''35'''  (1847)  pp. 319–326</TD></TR>
 +
<TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Vinogradov,  "Elements of number theory" , Dover, reprint  (1954)  (Translated from Russian)</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>

Revision as of 22:13, 30 November 2014

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

An element of a ring is divisible by another element if there exists a such that . One also says that divides and is said to be a multiple of , while is the divisor of . The divisibility of by is denoted by the symbol .

Any associative-commutative ring displays the following divisibility properties:

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

In an integral domain, elements and are simultaneously divisible by each other ( and ) if and only if they are associated, i.e. , where 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=18117
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