View source for Divisibility in rings

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 rings and algebras|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 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 ring]]s''. 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====
<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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