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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 $\mathfrak{a=bc}$ 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 $\mathfrak{p}$.

# 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 $\mathcal{O}$. 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 $\mathfrak{a=b}$.

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 $\mathfrak{e}$.

# 0.3 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 $\mathcal{O}^{*}\to\mathfrak{D}^{{\prime}}$, 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.

# References

- 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).

## Mathematics Subject Classification

11A51*no label found*13A05

*no label found*

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## Attached Articles

divisor theory and exponent valuations by pahio

any divisor is gcd of two principal divisors by pahio

divisor theory in finite extension by pahio

Chinese remainder theorem in terms of divisor theory by pahio

divisor as factor of principal divisor by pahio

image ideal of divisor by pahio

## 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!