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 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, $\mathfrak{e}$ 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 $\mathfrak{p}$ .

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 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. A divisibility $\alpha \mid \beta$ in $\mathcal{O}^*$ is valid iff the divisibility $(\alpha) \mid (\beta)$ is valid in $\mathfrak{D}$ .
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. If $\{\alpha\in\mathcal{O}\,\vdots\;\, \mathfrak{a} \mid \alpha\} = \{\beta\in\mathcal{O}\,\vdots\;\, \mathfrak{b} \mid \beta\}$ , then $\mathfrak{a = b}$ .
   

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 $\mathfrak{e}$ .

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 $\mathcal{O}^* \to \mathfrak{D}'$ , there is an isomorphism $\varphi\!:\, \mathfrak{D} \to \mathfrak{D}'$ such that $\varphi((\alpha)) = (\alpha)'$ always when the principal divisors $(\alpha)\in\mathfrak{D}$ and $(\alpha)'\in\mathfrak{D}'$ 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.

Bibliography

1

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

2

. . :. ``''. (1982).