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 prime element 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 presentation is unique up to the order of the prime factors $\mathfrak{p}_{i}$; 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 $\mathfrak{p}$.

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 divisor theory, 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 $\alpha\mid\beta$ 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 addition, 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 divisors and especially the divisors of the form $(\alpha)$, where  $\alpha\in\mathcal{O}^{*}$, principal divisors.  The prime elements of $\mathfrak{D}$ 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 $\mathcal{O}$ determine the unit divisor $\mathfrak{e}$.

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 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.