divisor theory and exponent valuations
A divisor theory^{} ${\mathcal{O}}^{*}\to \U0001d507$ of an integral domain^{} $\mathcal{O}$ determines via its prime divisors^{} a certain set $N$ of exponent valuations on the quotient field of $\mathcal{O}$. Assume to be known this set of exponents (http://planetmath.org/ExponentValuation2) ${\nu}_{\U0001d52d}$ corresponding the prime divisors $\U0001d52d$. There is a bijective^{} 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 $\U0001d507$ of all divisors^{} in question. The homomorphism^{} ${\mathcal{O}}^{*}\to \U0001d507$ is then defined by the condition
$\alpha \mapsto {\displaystyle \prod _{i}}\U0001d52d_{\U0001d526}{}^{{\nu}_{{\U0001d52d}_{i}}(\alpha )}=(\alpha ),\xc3\mathrm{\x82}\xc2\mathrm{\xb4}$  (1) 
since for any element $\alpha $ of ${\mathcal{O}}^{*}$ there exists only a finite number of exponents ${\nu}_{{\U0001d52d}_{i}}$ which do not vanish on $\alpha $ (corresponding the different prime divisor factors (http://planetmath.org/DivisibilityInRings) of the principal divisor $(\alpha )$).
One can take the concept of exponent as foundation for divisor theory:
Theorem. Let $\mathcal{O}$ 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 $\mathcal{O}$ iff the following three conditions are in :

•
For every $\alpha \in \mathcal{O}$ there is at most a finite number of exponents $\nu \in N$ such that $\nu (\alpha )\ne 0$.

•
An element $\alpha \in K$ belongs to $\mathcal{O}$ if and only if $\nu (\alpha )\geqq 0$ for each $\nu \in N$.

•
For any finite set^{} ${\nu}_{1},\mathrm{\dots},{\nu}_{n}$ of distinct exponents in $N$ and for the arbitrary set ${k}_{1},\mathrm{\dots},{k}_{n}$ of nonnegative integers, there exists an element $\alpha $ of $\mathcal{O}$ such that
$${\nu}_{1}(\alpha )={k}_{1},\mathrm{\dots},{\nu}_{n}(\alpha )={k}_{n}.$$
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 monoid^{} $\U0001d507$ that allows unique prime factorisation and that may be mapped bijectively onto $N$. The exponent in $N$ which corresponds to arbitrary prime element^{} $\U0001d52d$ is denoted by ${\nu}_{\U0001d52d}$. Then we obtain the homomorphism
$$\alpha \mapsto \prod _{\nu}{\U0001d52d}^{{\nu}_{\U0001d52d}(\alpha )}:=(\alpha )$$ 
which can be seen to satisfy all required properties for a divisor theory ${\mathcal{O}}^{*}\to \U0001d507$.
References
 1 S. Borewicz & I. Safarevic: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).
Title  divisor theory and exponent valuations 

Canonical name  DivisorTheoryAndExponentValuations 
Date of creation  20130322 17:59:34 
Last modified on  20130322 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 