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”).
iff [see the principal ideals].
0 is divisible by all elements of .
iff is a unit of .
All elements of are divisible by every unit of .
If then .
If then and .
If and then .
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|
|Date of creation||2015-05-06 15:18:14|
|Last modified on||2015-05-06 15:18:14|
|Last modified by||pahio (2872)|
|Defines||divisibility of ideals|