divisibility in rings
Let $(A,+,\cdot )$ be a commutative ring with a nonzero 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 $a\ne 0$, 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”).

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

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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 nonzero 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 $R$: $\U0001d51e\mid \U0001d51f\iff (\exists \U0001d52e)(\U0001d51f=\U0001d52e\U0001d51e)$. Cf. multiplication ring.
Title  divisibility in rings 
Canonical name  DivisibilityInRings 
Date of creation  20150506 15:18:14 
Last modified on  20150506 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 