# divisor theory and exponent valuations

A divisor theory$\mathcal{O}^{*}\to\mathfrak{D}$  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) $\nu_{\mathfrak{p}}$ corresponding the prime divisors $\mathfrak{p}$.  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 $\mathfrak{D}$ of all divisors in question. The homomorphism$\mathcal{O}^{*}\to\mathfrak{D}$  is then defined by the condition

 $\displaystyle\alpha\;\mapsto\;\prod_{i}\mathfrak{p_{i}}^{\nu_{\mathfrak{p}_{i}% }(\alpha)}=(\alpha),Â´$ (1)

since for any element $\alpha$ of $\mathcal{O}^{*}$ there exists only a finite number of exponents $\nu_{\mathfrak{p}_{i}}$ 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:

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  $\nu(\alpha)\neq 0$.

• 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$\nu_{1},\,\ldots,\,\nu_{n}$  of distinct exponents in $N$ and for the arbitrary set  $k_{1},\,\ldots,\,k_{n}$ of non-negative integers, there exists an element $\alpha$ of $\mathcal{O}$ such that

 $\nu_{1}(\alpha)=k_{1},\,\;\ldots,\,\;\nu_{n}(\alpha)=k_{n}.$

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 $\mathfrak{D}$ that allows unique prime factorisation and that may be mapped bijectively onto $N$.  The exponent in $N$ which corresponds to arbitrary prime element $\mathfrak{p}$ is denoted by $\nu_{\mathfrak{p}}$.  Then we obtain the homomorphism

 $\alpha\mapsto\prod_{\nu}\mathfrak{p}^{\nu_{\mathfrak{p}}(\alpha)}:=(\alpha)$

which can be seen to satisfy all required properties for a divisor theory  $\mathcal{O}^{*}\to\mathfrak{D}$.

## References

• 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
Title divisor theory and exponent valuations DivisorTheoryAndExponentValuations 2013-03-22 17:59:34 2013-03-22 17:59:34 pahio (2872) pahio (2872) 7 pahio (2872) Topic msc 13A18 msc 12J20 msc 13A05 divisors and exponents ExponentValuation2 ImplicationsOfHavingDivisorTheory