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Divisibility in rings

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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://www.encyclopediaofmath.org/index.php?title=Divisibility_in_rings&oldid=41964
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