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 $\nu_{\mathfrak{p}}$ 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 structure of the free monoid $\mathfrak{D}$ of all divisors in question. The homomorphism $\mathcal{O}^* \to \mathfrak{D}$ is then defined by the condition

(1)

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 of $K$ . The exponents in $N$ determine, as in (1), a divisor theory of $\mathcal{O}$ iff the following three conditions are in force:

• 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 $\mathcal{O}^* \to \mathfrak{D}$ .

Bibliography

1

S. BOREWICZ & I. SAFAREVIC: Zahlentheorie. Birkhäuser Verlag. Basel und Stuttgart (1966).