# implications of having divisor theory

The existence of a divisor theory restricts strongly the of an integral domain^{}, as is seen from the following propositions.

Proposition 1. An integral domain $\mathcal{O}$ which has a divisor theory ${\mathcal{O}}^{*}\to \U0001d507$, is integrally closed^{} in its quotient field.

Proof. Let $\xi $ be an element of the quotient field of $\mathcal{O}$ which is integral over $\mathcal{O}$. Then $\xi $ satisfies an equation

${\xi}^{n}+{\alpha}_{1}{\xi}^{n-1}+\mathrm{\dots}+{\alpha}_{n}=0$ | (1) |

where ${\alpha}_{1},\mathrm{\dots},{\alpha}_{n}\in \mathcal{O}$. Now, we can write $\xi ={\displaystyle \frac{\varkappa}{\lambda}}$ with $\varkappa ,\lambda \in \mathcal{O}$, whence (1) may be written

${\varkappa}^{n}=-{\alpha}_{1}\lambda {\varkappa}^{n-1}-{\alpha}_{2}{\lambda}^{2}{\varkappa}^{n-2}-\mathrm{\dots}-{\alpha}_{n}{\lambda}^{n}.$ | (2) |

Let us make the antithesis that $\xi $ does not belong to $\mathcal{O}$ itself. Then $\lambda \nmid \varkappa $ and therefore we have for the corresponding principal divisors $(\lambda )\nmid (\varkappa )$. We infer that there is a prime divisor^{} factor $\U0001d52d$ of $(\lambda )$ and an integer $k\geqq 0$ such that

$${\U0001d52d}^{k}\mid (\varkappa ),{\U0001d52d}^{k+1}\nmid (\varkappa ),{\U0001d52d}^{k+1}\mid (\lambda ).$$ |

By the condition 2 of the definition of divisor theory (http://planetmath.org/DivisorTheory), the right hand side of the equation (2) is divisible by

$${\U0001d52d}^{(k+1)+(n-1)k}={\U0001d52d}^{kn+1}.$$ |

On the other side, the highest power of $\U0001d52d$, by which the divisor^{} $({\varkappa}^{n})$ is divisible, is ${\U0001d52d}^{kn}$. Accordingly, the different sides of (2) show different divisibility by powers of $\U0001d52d$. This contradictory situation means that the antithesis was wrong and thus the proposition has been proven.

Proposition 2. When an integral domain $\mathcal{O}$ has a divisor theory ${\mathcal{O}}^{*}\to \U0001d507$, then each element of ${\mathcal{O}}^{*}$ has only a finite number of non-associated (http://planetmath.org/Associates^{}) factors (http://planetmath.org/DivisibilityInRings).

Proof. Let $\xi $ be an arbitrary non-zero element of $\mathcal{O}$. We form the prime factor^{} presentation^{} of the corresponding principal divisor $(\xi )$:

$$(\xi )={\U0001d52d}_{1}{\U0001d52d}_{2}\mathrm{\cdots}{\U0001d52d}_{r}$$ |

This is unique up to the ordering of the factors; $r\geqq 0$. Then we form of the prime divisors ${\U0001d52d}_{i}$ all products having $k$ factors ($0\leqq k\leqq r$) and choose from the products those which are principal divisors. Thus we obtain a set of factors of $(\xi )$ containing at most $\left({\displaystyle \genfrac{}{}{0pt}{}{r}{k}}\right)$ elements. All different principal divisor factors of $(\xi )$ are gotten, as $k$ runs all integers from 0 to $r$, and their number is at most equal to

$$\sum _{k=0}^{r}\left(\genfrac{}{}{0pt}{}{r}{k}\right)={2}^{r}$$ |

(see 5. in the binomial coefficients^{}). To every principal divisor , there corresponds a class (http://planetmath.org/EquivalenceClass) of associate factors of $\xi $, and the elements of distinct classes (http://planetmath.org/EquivalenceClass) are non-associates. Since $\xi $ has not other factors, the number of its non-associated factors is at most ${2}^{r}$.

Title | implications of having divisor theory |
---|---|

Canonical name | ImplicationsOfHavingDivisorTheory |

Date of creation | 2013-03-22 17:59:13 |

Last modified on | 2013-03-22 17:59:13 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 6 |

Author | pahio (2872) |

Entry type | Topic |

Classification | msc 11A51 |

Classification | msc 13A05 |

Synonym | properties of rings having a divisor theory |

Related topic | DivisorTheoryAndExponentValuations |