PlanetMath: divisibility in rings

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divisibility in rings

(Definition)

Let $(A,\,+,\,\cdot)$ be a commutative ring with a non-zero unity 1. If and are two elements of and if there is an element of such that , then is said to be divisible by ; it may be denoted by $a\mid b$ . (If has no zero divisors and $a \neq 0$ , then is uniquely determined.)

Properties

$a\mid b$ iff $(b)\subseteq (a)$ [see the principal ideals].
Divisibility is a reflexive and transitive relation in .
0 is divisible by all elements of .
$a\mid 1$ iff is a unit of .
All elements of are divisible by every unit of .
If $a\mid b$ then $a\mid bc$ and $ac\mid bc$ .
If $a\mid b$ and $a\mid c$ then $a\mid b+c$ .
If $a\mid b$ and $a\nmid c$ then $a\nmid b+c$ .

Note. The divisibility can be similarly defined if $(A,\,+,\,\cdot)$ is only a semiring, and it also has the above properties except the first. This concerns especially the case that we have a ring with non-zero unity and is the set of the ideals of (see the ideal multiplication laws). Thus one may speak of the divisibility of ideals in : $\mathfrak{a\mid b\,\,\Leftrightarrow\,\, (\exists q)\,(b = qa)}$ . Cf. multiplication ring.

"divisibility in rings" is owned by pahio.

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Also defines:

divisible, divisibility, divisibility of ideals

Keywords:

divide, divisor, factor

This object's parent.

Attachments:: divisibility by product (Theorem) by pahio
strict divisibility (Definition) by pahio
divisor theory (Definition) by pahio

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Cross-references: multiplication ring, ideal multiplication laws, ideals, ring, properties, semiring, unit, transitive relation, Reflexive, principal ideals, iff, zero divisors, non-zero unity, commutative ring
There are 144 references to this entry.

This is version 14 of divisibility in rings, born on 2004-10-08, modified 2007-03-19.
Object id is 6322, canonical name is DivisibilityInRings.
Accessed 5970 times total.

Classification:

AMS MSC:	11A51 (Number theory :: Elementary number theory :: Factorization; primality)
	13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)

Pending Errata and Addenda

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