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

When $b$ is divisible by $a$, $a$ is said to be a divisor or factor of $b$. On the other hand, $b$ is not said to be a multiple of $a$ except in the case that $A$ is the ring $\mathbb{Z}$ of the integers. In some languages, e.g. in the Finnish, $b$ has a name which could be approximately be translated as ‘containant’: $b$ is a containant of $a$ (“$b$ on $a$:n sisältäjä”).

Properties

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$a\mid b$ iff $(b)\subseteq(a)$ [see the principal ideals].

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Divisibility is a reflexive and transitive relation in $A$.

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0 is divisible by all elements of $A$.

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$a\mid 1$ iff $a$ is a unit of $A$.

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All elements of $A$ are divisible by every unit of $A$.

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

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If $a\mid b$ then $a\mid bc$ and $ac\mid bc$.

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If $a\mid b$ and $a\mid c$ then $a\mid b\!+\!c$.

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