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.)

Properties

• $a\mid b$ ; iff $(b)\subseteq (a)$ , [see the principal ideals].
• Divisibility is a reflexive and transitive relation in $A$
• 0 is divisible by all elements of $A$
• $a\mid 1$ ; iff $a$ is a unit of $A$
• All elements of $A$ are divisible by every unit of $A$
• 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 $R$ 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.