divisibility in rings

Let  (A,+,)  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  ab.  (If A has no zero divisorsMathworldPlanetmath and  a0,  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 of the integers.  In some languagesPlanetmathPlanetmath, 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”).

  • ab  iff  (b)(a)   [see the principal idealsMathworldPlanetmathPlanetmathPlanetmathPlanetmath].

  • Divisibility is a reflexiveMathworldPlanetmathPlanetmathPlanetmath and transitive relation in A.

  • 0 is divisible by all elements of A.

  • a1  iff  a is a unit of A.

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

  • If  ab  then  anbn(n=1, 2,).

  • If  ab  then  abc  and  acbc.

  • If  ab  and  ac  then  ab+c.

  • If  ab  and  ac  then  ab+c.

Note.  The divisibility can be similarly defined if  (A,+,)  is only a semiringMathworldPlanetmath; 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:  𝔞𝔟(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 IrreduciblePlanetmathPlanetmathPlanetmath
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