implications of having divisor theory

The existence of a divisor theory restricts strongly the of an integral domainMathworldPlanetmath, as is seen from the following propositions.

Proposition 1.  An integral domain 𝒪 which has a divisor theory  𝒪*→𝔇,  is integrally closedMathworldPlanetmath in its quotient field.

Proof.  Let ξ be an element of the quotient field of 𝒪 which is integral over 𝒪.  Then ξ satisfies an equation

ξn+α1⁢ξn-1+…+αn=0 (1)

where  α1,…,αn∈𝒪.  Now, we can write  ξ=ϰλ  with  ϰ,λ∈𝒪,  whence (1) may be written

ϰn=-α1⁢λ⁢ϰn-1-α2⁢λ2⁢ϰn-2-…-αn⁢λn. (2)

Let us make the antithesis that ξ does not belong to 𝒪 itself.  Then  λ∤ϰ  and therefore we have for the corresponding principal divisors  (λ)∤(ϰ).  We infer that there is a prime divisorPlanetmathPlanetmath factor 𝔭 of (λ) and an integer k≧0 such that


By the condition 2 of the definition of divisor theory (, the right hand side of the equation (2) is divisible by


On the other side, the highest power of 𝔭, by which the divisorMathworldPlanetmathPlanetmathPlanetmath (ϰn) is divisible, is 𝔭k⁢n.  Accordingly, the different sides of (2) show different divisibility by powers of 𝔭.  This contradictory situation means that the antithesis was wrong and thus the proposition has been proven.

Proposition 2.  When an integral domain 𝒪 has a divisor theory  𝒪*→𝔇,  then each element of 𝒪* has only a finite number of non-associated ( factors (

Proof.  Let ξ be an arbitrary non-zero element of 𝒪.  We form the prime factorMathworldPlanetmath presentationMathworldPlanetmathPlanetmath of the corresponding principal divisor (ξ):


This is unique up to the ordering of the factors;  r≧0.  Then we form of the prime divisors 𝔭i all products having k factors (0≦k≦r) and choose from the products those which are principal divisors.  Thus we obtain a set of factors of (ξ) containing at most (rk) elements.  All different principal divisor factors of (ξ) are gotten, as k runs all integers from 0 to r, and their number is at most equal to


(see 5. in the binomial coefficientsMathworldPlanetmath).  To every principal divisor , there corresponds a class ( of associate factors of ξ, and the elements of distinct classes ( are non-associates.  Since ξ has not other factors, the number of its non-associated factors is at most 2r.

Title implications of having divisor theory
Canonical name ImplicationsOfHavingDivisorTheory
Date of creation 2013-03-22 17:59:13
Last modified on 2013-03-22 17:59:13
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 6
Author pahio (2872)
Entry type Topic
Classification msc 11A51
Classification msc 13A05
Synonym properties of rings having a divisor theory
Related topic DivisorTheoryAndExponentValuations