PlanetMath: divisor theory

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divisor theory

(Definition)

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:

A divisibility $\alpha \mid \beta$ in $\mathcal{O}^*$ is valid iff the divisibility $(\alpha) \mid (\beta)$ is valid in $\mathfrak{D}$ .
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 $\mathfrak{D}$ ).
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 $\mathfrak{e}$ .

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: М. М. Постников: Введение в теорию алгебраических чисел. Издательство ``Наука''. Москва(1982).

"divisor theory" is owned by pahio.

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Also defines:

divisor, prime divisor, principal divisor, unit divisor

Keywords:

prime factorization

This object's parent.

Attachments:: implications of having divisor theory (Topic) by pahio
divisor theory and exponent valuations (Topic) by pahio
any divisor is gcd of two principal divisors (Theorem) by pahio
divisor theory in finite extension (Theorem) by pahio
Chinese remainder theorem in terms of divisor theory (Theorem) by pahio
divisor as factor of principal divisor (Theorem) by pahio
image ideal of divisor (Theorem) by pahio

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Cross-references: isomorphism, units, associates, properties, homomorphism, multiplication, identity, integral domain, greatest common factor, free monoid, product, finite, prime, monoid, prime element, unity, iff, divisible, divisibility, commutative monoid
There are 82 references to this entry.

This is version 11 of divisor theory, born on 2008-04-10, modified 2008-05-08.
Object id is 10494, canonical name is DivisorTheory.
Accessed 542 times total.

Classification:

AMS MSC:	11A51 (Number theory :: Elementary number theory :: Factorization; primality)
	13A05 (Commutative rings and algebras :: General commutative ring theory :: Divisibility)

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