divisor theory and exponent valuations

A divisor theoryMathworldPlanetmath𝒪*𝔇  of an integral domainMathworldPlanetmath 𝒪 determines via its prime divisorsPlanetmathPlanetmath a certain set N of exponent valuations on the quotient field of 𝒪.  Assume to be known this set of exponents (http://planetmath.org/ExponentValuation2) ν𝔭 corresponding the prime divisors 𝔭.  There is a bijectiveMathworldPlanetmathPlanetmath correspondence between the elements of N and of the set of all prime divisors.  The set of the prime divisors determines completely the of the free monoid 𝔇 of all divisorsMathworldPlanetmathPlanetmathPlanetmath in question. The homomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath𝒪*𝔇  is then defined by the condition

αi𝔭𝔦ν𝔭i(α)=(α),´ (1)

since for any element α of 𝒪* there exists only a finite number of exponents ν𝔭i which do not vanish on α (corresponding the different prime divisor factors (http://planetmath.org/DivisibilityInRings) of the principal divisor (α)).

One can take the concept of exponent as foundation for divisor theory:

Theorem.  Let 𝒪 be an integral domain with quotient field K and N a given set of exponents (http://planetmath.org/ExponentValuation2) of K.  The exponents in N determine, as in (1), a divisor theory of 𝒪 iff the following three conditions are in :

  • For every  α𝒪  there is at most a finite number of exponents  νN  such that  ν(α)0.

  • An element  αK  belongs to 𝒪 if and only if  ν(α)0  for each  νN.

  • For any finite setMathworldPlanetmathν1,,νn  of distinct exponents in N and for the arbitrary set  k1,,kn of non-negative integers, there exists an element α of 𝒪 such that


For the proof of the theorem, we mention only how to construct the divisors when we have the exponent set N fulfilling the three conditions of the theorem.  We choose a commutative monoidPlanetmathPlanetmath 𝔇 that allows unique prime factorisation and that may be mapped bijectively onto N.  The exponent in N which corresponds to arbitrary prime elementMathworldPlanetmath 𝔭 is denoted by ν𝔭.  Then we obtain the homomorphism


which can be seen to satisfy all required properties for a divisor theory  𝒪*𝔇.


  • 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title divisor theory and exponent valuations
Canonical name DivisorTheoryAndExponentValuations
Date of creation 2013-03-22 17:59:34
Last modified on 2013-03-22 17:59:34
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Topic
Classification msc 13A18
Classification msc 12J20
Classification msc 13A05
Synonym divisors and exponents
Related topic ExponentValuation2
Related topic ImplicationsOfHavingDivisorTheory