# any divisor is gcd of two principal divisors

Using the exponent valuations, one can easily prove the

Theorem. In any divisor theory, each divisor^{} is the greatest common divisor^{} of two principal divisors.

Proof. Let ${\mathcal{O}}^{*}\to \U0001d507$ be a divisor theory and $\U0001d521$ an arbitrary divisor in $\U0001d507$. We may suppose that $\U0001d521$ is not a principal divisor (if $\U0001d507$ contains exclusively principal divisors, then $\U0001d521=\mathrm{gcd}(\U0001d521,\U0001d521)$ and the proof is ready). Let

$$\U0001d521=\prod _{i=1}^{r}{\U0001d52d}_{i}^{{k}_{i}}$$ |

where the ${\U0001d52d}_{i}$’s are pairwise distinct prime divisors and every ${k}_{i}>0$. Then third condition in the theorem concerning divisors and exponents allows to choose an element $\alpha $ of the ring $\mathcal{O}$ such that

$${\nu}_{{\U0001d52d}_{1}}(\alpha )={k}_{1},\mathrm{\dots},{\nu}_{{\U0001d52d}_{r}}(\alpha )={k}_{r}.$$ |

Let the principal divisor corresponding to $\alpha $ be

$$(\alpha )=\prod _{i=1}^{r}{\U0001d52d}_{i}^{{k}_{i}}\prod _{j=1}^{s}{\U0001d52e}_{j}^{{l}_{j}}=\U0001d521{\U0001d521}^{\prime},$$ |

where the prime divisors ${\U0001d52e}_{j}$ are pairwise different among themselves and with the divisors ${\U0001d52d}_{i}$. We can then choose another element $\beta $ of $\mathcal{O}$ such that

$${\nu}_{{\U0001d52d}_{1}}(\beta )={k}_{1},\mathrm{\dots},{\nu}_{{\U0001d52d}_{r}}(\beta )={k}_{r},{\nu}_{{\U0001d52e}_{1}}(\beta )=\mathrm{\dots}={\nu}_{{\U0001d52e}_{s}}(\beta )=0.$$ |

Then we have $(\beta )=\U0001d521{\U0001d521}^{\prime \prime}$, where ${\U0001d521}^{\prime \prime}\in \U0001d507$ and

$$\mathrm{gcd}({\U0001d521}^{\prime},{\U0001d521}^{\prime \prime})={\U0001d52e}^{0}\mathrm{\cdots}{\U0001d52e}^{0}=\U0001d522=(1).$$ |

The gcd of the principal divisors $(\alpha )$ and $(\beta )$ is apparently $\U0001d521$, whence the proof is settled.

Title | any divisor is gcd of two principal divisors |
---|---|

Canonical name | AnyDivisorIsGcdOfTwoPrincipalDivisors |

Date of creation | 2013-03-22 17:59:37 |

Last modified on | 2013-03-22 17:59:37 |

Owner | pahio (2872) |

Last modified by | pahio (2872) |

Numerical id | 5 |

Author | pahio (2872) |

Entry type | Theorem |

Classification | msc 13A05 |

Classification | msc 13A18 |

Classification | msc 12J20 |

Related topic | TwoGeneratorProperty |

Related topic | SumOfIdeals |