divisor theory

0.1 Divisibility in a monoid

In a commutative monoid $𝔇$, one can speak of divisibility: its element $𝔞$ is divisible by its element $𝔟$, iff  $𝔞=𝔟𝔠$  where  $𝔠\in 𝔇$.  An element $𝔭$ of $𝔇$, distinct from the unity $𝔢$ of $𝔇$, is called a prime element of $𝔇$, when $𝔭$ is divisible only by itself and $𝔢$.  The monoid $𝔇$ has a unique prime factorisation, if every element $𝔞$ of $𝔇$ can be presented as a finite product  $𝔞={𝔭}_1{𝔭}_2\mathrm{\cdots }{𝔭}_r$  of prime elements and this is unique up to the ${𝔭}_i$; then we may say that $𝔇$ is a free monoid on the set of its prime elements.

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

0.2 Divisor theory of an integral domain

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

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

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

3. If  $\{\alpha \in 𝒪\mathrm{⋮}𝔞\mid \alpha \}=\{\beta \in 𝒪\mathrm{⋮}𝔟\mid \beta \}$,  then  $𝔞=𝔟$.
   

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

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 $𝒪$ determine the unit divisor $𝔢$.

0.3 Uniqueness theorems

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

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

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

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

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