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Ideal Unity Commutative Ring Reflexive Non Degenerate Sesquilinear Ring Biconditional Property Transitive Zero Divisor Principal Ideal Multiplication Ring Semiring Ideal Multiplication Laws

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Divisibility In Rings

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 $a^n\mid b^n \;\; (n = 1,\,2,\,\ldots)$ .
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.

1	Reflexive Non Degenerate Sesquilinear
1	Property
1	Biconditional
1	Commutative Ring
1	Ring
1	Unity
1	Ideal
2	Transitive
3	Zero Divisor
3	Principal Ideal
5	Semiring
6	Ideal Multiplication Laws
6	Multiplication Ring