# divisor theory

## 0.1 Divisibility in a monoid

In a commutative monoid $\mathfrak{D}$, one can speak of divisibility: its element $\mathfrak{a}$ is divisible by its element $\mathfrak{b}$, iff  $\mathfrak{a=bc}$  where  $\mathfrak{c}\in\mathfrak{D}$.  An element $\mathfrak{p}$ of $\mathfrak{D}$, distinct from the unity $\mathfrak{e}$ of $\mathfrak{D}$, is called a of $\mathfrak{D}$, when $\mathfrak{p}$ is divisible only by itself and $\mathfrak{e}$.  The monoid $\mathfrak{D}$ has a unique prime factorisation, if every element $\mathfrak{a}$ of $\mathfrak{D}$ can be presented as a finite product$\mathfrak{a=p}_{1}\mathfrak{p}_{2}\cdots\mathfrak{p}_{r}$  of prime elements and this is unique up to the $\mathfrak{p}_{i}$; then we may say that $\mathfrak{D}$ is a free monoid on the set of its prime elements.

If the monoid $\mathfrak{D}$ has a unique prime factorisation, $\mathfrak{e}$ is divisible only by itself.  Two elements of $\mathfrak{D}$ have always a greatest common factor.  If a product $\mathfrak{ab}$ is divisible by a prime element $\mathfrak{p}$, then at least one of $\mathfrak{a}$ and $\mathfrak{b}$ is divisible by $\mathfrak{p}$.

## 0.2 Divisor theory of an integral domain

Let $\mathcal{O}$ be an integral domain and $\mathcal{O}^{*}$ the set of its non-zero elements; this set forms a commutative monoid (with identity 1) with respect to the multiplication of $\mathcal{O}$.  We say that the integral domain $\mathcal{O}$ has a , if there is a commutative monoid $\mathfrak{D}$ with unique prime factorisation and a homomorphism$\alpha\mapsto(\alpha)$  from the monoid $\mathcal{O}^{*}$ into the monoid $\mathfrak{D}$, such that the following three properties are true:

1. 1.

A divisibility (http://planetmath.org/DivisibilityInRings) $\alpha\mid\beta$ in $\mathcal{O}^{*}$ is valid iff the divisibility $(\alpha)\mid(\beta)$ is valid in $\mathfrak{D}$.

2. 2.

If the elements $\alpha$ and $\beta$ of $\mathcal{O}^{*}$ are divisible by an element $\mathfrak{c}$ of $\mathfrak{D}$, then also $\alpha\pm\beta$ are divisible by $\mathfrak{c}$  (‘‘$\mathfrak{c}\mid\alpha$’’  means that  $\mathfrak{c}\mid(\alpha)$;  in , 0 is divisible by every element of $\mathfrak{D}$).

3. 3.

If  $\{\alpha\in\mathcal{O}\,\vdots\;\,\mathfrak{a}\mid\alpha\}=\{\beta\in\mathcal{% O}\,\vdots\;\,\mathfrak{b}\mid\beta\}$,  then  $\mathfrak{a=b}$.

A divisor theory of $\mathcal{O}$ is denoted by  $\mathcal{O}^{*}\to\mathfrak{D}$.  The elements of $\mathfrak{D}$ are called and especially the divisors of the form $(\alpha)$, where  $\alpha\in\mathcal{O}^{*}$, principal divisors.  The prime elements of $\mathfrak{D}$ are .

By 1, it is easily seen that two principal divisors $(\alpha)$ and $(\beta)$ are equal iff the elements $\alpha$ and $\beta$ are associates of each other.  Especially, the units of $\mathcal{O}$ determine the unit divisor $\mathfrak{e}$.

## 0.3 Uniqueness theorems

Theorem 1.  An integral domain $\mathcal{O}$ has at most one divisor theory.  In other words, for any pair of divisor theories  $\mathcal{O}^{*}\to\mathfrak{D}$  and  $\mathcal{O}^{*}\to\mathfrak{D}^{\prime}$, there is an isomorphism$\varphi\!:\,\mathfrak{D}\to\mathfrak{D}^{\prime}$  such that  $\varphi((\alpha))=(\alpha)^{\prime}$  always when the principal divisors  $(\alpha)\in\mathfrak{D}$  and  $(\alpha)^{\prime}\in\mathfrak{D}^{\prime}$  correspond to the same element $\alpha$ of $\mathcal{O}^{*}$.

Theorem 2.  An integral domain $\mathcal{O}$ is a unique factorisation domain (http://planetmath.org/UFD) if and only if $\mathcal{O}$ has a divisor theory  $\mathcal{O}^{*}\to\mathfrak{D}$  in which all divisors are principal divisors.

Theorem 3.  If the divisor theory  $\mathcal{O}^{*}\to\mathfrak{D}$  comprises only a finite number of prime divisors, then $\mathcal{O}$ is a unique factorisation domain.

The proofs of those theorems are found in [1], which is available also in Russian (original), English and French.

## References

• 1 S. Borewicz & I. Safarevic: Zahlentheorie.  Birkhäuser Verlag. Basel und Stuttgart (1966).
• 2 М. М. Постников: Введение  в  теорию  алгебраических  чисел.  Издательство  ‘‘Наука’’. Москва (1982).
 Title divisor theory Canonical name DivisorTheory Date of creation 2013-03-22 17:59:03 Last modified on 2013-03-22 17:59:03 Owner pahio (2872) Last modified by pahio (2872) Numerical id 17 Author pahio (2872) Entry type Definition Classification msc 11A51 Classification msc 13A05 Related topic UniqueFactorizationAndIdealsInRingOfIntegers Related topic IdealDecompositionInDedekindDomain Related topic EisensteinCriterionInTermsOfDivisorTheory Related topic DivisorsInBaseFieldAndFiniteExtensionField Related topic ExponentOfField Related topic ExponentValuation2 Related topic DedekindDomainsWithFinitelyManyPrimesArePIDs Defines divisor Defines prime divisor Defines principal divisor Defines unit divisor