divisibility in rings
Let be a commutative ring with a non-zero
unity 1. If and are two elements of and if there
is an element of such that , then is
said to be divisible by ; it may be denoted by
. (If has no zero divisors and ,
then is uniquely determined.)
When is divisible by , is said to be a
divisor or
factor (http://planetmath.org/DivisibilityInRings)
of . On the other hand, is not said to be
a multiple of except in the case that is the
ring of the integers. In some languages, e.g. in
the Finnish, has a name which could be approximately be
translated as ‘containant’: is a containant
of (“ on :n sisältäjä”).
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•
iff [see the principal ideals
].
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•
Divisibility is a reflexive
and transitive relation in .
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•
0 is divisible by all elements of .
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iff is a unit of .
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All elements of are divisible by every unit of .
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If then .
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If then and .
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If and then .
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If and then .
Note. The divisibility can be similarly defined if
is only a semiring; then it also has the
above properties except the first. This concerns especially
the case that we have a ring with non-zero unity and is
the set of the ideals of (see the ideal multiplication laws).
Thus one may speak of the divisibility of ideals in
: . 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 |