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.