divisibility in rings

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

When $b$ is divisible by $a$, $a$ is said to be adivisor orfactor (http://planetmath.org/DivisibilityInRings)of $b$.  On the other hand, $b$ is not said to bea multiple of $a$ except in the case that $A$ is thering $\mathbb{Z}$ of the integers.  In some languages, e.g. inthe Finnish, $b$ has a name which could be approximately betranslated as ‘containant’: $b$ is a containantof $a$ (“$b$ on $a$:n sisältäjä”).

• $a\mid b$  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  $a^n\mid b^n(n=1,\mathrm{\hspace{0.17em}2},\mathrm{\dots })$.

• 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; then it also has theabove properties except the first.  This concerns especiallythe case that we have a ring $R$ with non-zero unity and $A$ isthe set of the ideals of $R$ (see the ideal multiplication laws). Thus one may speak of the divisibility of ideals in$R$:  $𝔞\mid 𝔟\iff (\exists 𝔮)(𝔟=𝔮𝔞)$.  Cf. multiplication ring.