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Homedivisor theory

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# divisor theory

# 0.1 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 โ$\mathfrak{a=bc}$โ whereโ $\mathfrak{c}\in\mathfrak{D}$.โ 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 $\mathfrak{e}$.โ The monoid $\mathfrak{D}$ has a unique prime factorisation, if every element $\mathfrak{a}$ of $\mathfrak{D}$ can be presented as a finite product โ$\mathfrak{a=p}_{1}\mathfrak{p}_{2}\cdots\mathfrak{p}_{r}$โ 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.โ Two elements of $\mathfrak{D}$ have always a greatest common factor.โ 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 $\mathfrak{p}$.

# 0.2 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 $\mathcal{O}$.โ 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 โ $\alpha\mapsto(\alpha)$โ 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 $\mathfrak{c}$โ (โโ$\mathfrak{c}\mid\alpha$โโโ means thatโ $\mathfrak{c}\mid(\alpha)$;โ in addition, 0 is divisible by every element of $\mathfrak{D}$).

3. Ifโ $\{\alpha\in\mathcal{O}\,\vdots\;\,\mathfrak{a}\mid\alpha\}=\{\beta\in\mathcal{% O}\,\vdots\;\,\mathfrak{b}\mid\beta\}$,โ thenโ $\mathfrak{a=b}$.

A divisor theory of $\mathcal{O}$ is denoted byโ $\mathcal{O}^{*}\to\mathfrak{D}$.โ The elements of $\mathfrak{D}$ are called divisors and especially the divisors of the form $(\alpha)$, whereโ $\alpha\in\mathcal{O}^{*}$, principal divisors.โ 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.โ Especially, the units of $\mathcal{O}$ determine the unit divisor $\mathfrak{e}$.

# 0.3 Uniqueness theorems

Theorem 1.โ An integral domain $\mathcal{O}$ has at most one divisor theory.โ In other words, for any pair of divisor theoriesโ $\mathcal{O}^{*}\to\mathfrak{D}$โ andโ $\mathcal{O}^{*}\to\mathfrak{D}^{{\prime}}$, there is an isomorphism โ$\varphi\!:\,\mathfrak{D}\to\mathfrak{D}^{{\prime}}$โ such thatโ $\varphi((\alpha))=(\alpha)^{{\prime}}$โ always when the principal divisorsโ $(\alpha)\in\mathfrak{D}$โ andโ $(\alpha)^{{\prime}}\in\mathfrak{D}^{{\prime}}$โ correspond to the same element $\alpha$ of $\mathcal{O}^{*}$.

Theorem 2.โ An integral domain $\mathcal{O}$ is a unique factorisation domain if and only if $\mathcal{O}$
has a divisor theoryโ $\mathcal{O}^{*}\to\mathfrak{D}$โ in which all divisors are principal divisors.

Theorem 3.โ If the divisor theoryโ $\mathcal{O}^{*}\to\mathfrak{D}$โ 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.

# References

- 1 S. Borewicz & I. Safarevic: Zahlentheorie.โ Birkhรคuser Verlag. Basel und Stuttgart (1966).
- 2 ะ. ะ. ะะพััะฝะธะบะพะฒ: ะะฒะตะดะตะฝะธะตโ ะฒโ ัะตะพัะธัโ ะฐะปะณะตะฑัะฐะธัะตัะบะธั โ ัะธัะตะป. โะะทะดะฐัะตะปัััะฒะพโ โโะะฐัะบะฐโโ. ะะพัะบะฒะฐโ(1982).

## Mathematics Subject Classification

11A51*no label found*13A05

*no label found*

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## Comments

## examples of divisor theory

Divisor theory looks very interesting. Do you have any examples of domains having a divisor theory which aren't PIDs or Dedekind domains?

Z[X] is a UFD, but not a PID, so prime factorization in Z[X] is not a valid divisor theory. Does it have one?

## Re: examples of divisor theory

Dear gel,

I cannot answer to your question -- about forty years have run after I studied the divisors. I took the material of my divisor entries from my old study work, from Postnikov's little book and from Borewicz--Shafarevic. Perhaps in some exercises of the latter, you can find some examples you want.

(Cf. http://planetmath.org/encyclopedia/ImageIdealOfDivisor.html)

Regards,

Jussi

## Re: examples of divisor theory

in fact, why isn't factorization in Z[X] a divisor theory? It seems to satisfy the requirements 1-3 listed in this entry, but all divisors are principal, which would contradict Theorem 2 that such domains are PIDs. Did I miss something here?

## Re: examples of divisor theory

Theorem 2 says that such domains are UFD's, not PID's.

## Re: examples of divisor theory

Sorry, I misread it. So factorization in Z[X] is a valid divisor theory.

## Re: examples of divisor theory

ok, I don't have those books, but now my confusion over Z[X] is sorted out, I think R[X] for R the ring of integers in a number field will have a division algebra but isnt a UFD or Dedekind domain in all cases.

## Re: examples of divisor theory

You certainly find the books at library. I recommend Borewicz--Shafarevic!