# divisibility in rings

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 (http://planetmath.org/DivisibilityInRings) 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”).

• $a\mid b$  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; 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.

 Title divisibility in rings Canonical name DivisibilityInRings Date of creation 2015-05-06 15:18:14 Last modified on 2015-05-06 15:18:14 Owner pahio (2872) Last modified by pahio (2872) Numerical id 24 Author pahio (2872) Entry type Definition Classification msc 13A05 Classification msc 11A51 Related topic PrimeElement Related topic Irreducible Related topic GroupOfUnits Related topic DivisibilityByPrimeNumber Related topic GcdDomain Related topic CorollaryOfBezoutsLemma Related topic ExistenceAndUniquenessOfTheGcdOfTwoIntegers Related topic MultiplicationRing Related topic IdealDecompositionInDedekindDomain Related topic IdealMultiplicationLaws Related topic UnityPlusNilpotentIsUnit Defines divisible Defines divisibility Defines divisibility of ideals