# example of ring which is not a UFD

###### Example 1.

We define a ring $R=\mathbb{Z}[\sqrt{-5}]=\{n+m\sqrt{-5}:n,m\in \mathbb{Z}\}$ with addition and multiplication inherited from $\u2102$ (notice that $R$ is the ring of integers^{} of the quadratic number field $\mathbb{Q}(\sqrt{-5})$). Notice that the only units (http://planetmath.org/UnitsOfQuadraticFields) of $R$ are ${R}^{\times}=\{\pm 1\}$. Then:

$6=2\cdot 3=(1+\sqrt{-5})\cdot (1-\sqrt{-5}).$ | (1) |

Moreover, $2,\mathrm{\hspace{0.25em}3},\mathrm{\hspace{0.25em}1}+\sqrt{-5}$ and $1-\sqrt{-5}$ are irreducible elements^{} of $R$ and they are not associates^{} (to see this, one can compare the norm of every element). Therefore, $R$ is not a UFD.

However, the ideals of $R$ factor (http://planetmath.org/DivisibilityInRings) uniquely into prime ideals^{}. For example:

$$(6)={(2,1+\sqrt{-5})}^{2}\cdot (3,1+\sqrt{-5})\cdot (3,1-\sqrt{-5})$$ |

where $\U0001d513=(2,1+\sqrt{-5})$, $\U0001d514=(3,1+\sqrt{-5})$, and $\overline{\U0001d514}=(3,1-\sqrt{-5})$ are all prime ideals (see prime ideal decomposition of quadratic extensions of $\mathbb{Q}$ (http://planetmath.org/PrimeIdealDecompositionInQuadraticExtensionsOfMathbbQ)). Notice that:

$${\U0001d513}^{2}=(2),\U0001d514\cdot \overline{\U0001d514}=(3),\U0001d513\cdot \U0001d514=(1+\sqrt{-5}),\U0001d513\cdot \overline{\U0001d514}=(1-\sqrt{-5}).$$ |

Thus, Eq. (1) above is the outcome of different rearrangements of the product of prime ideals:

$$(6)={\U0001d513}^{2}\cdot (\U0001d514\cdot \overline{\U0001d514})=(\U0001d513\cdot \U0001d514)\cdot (\U0001d513\cdot \overline{\U0001d514}).$$ |

Notice also that if $\U0001d513$ was a principal ideal^{} then there would be an element $\alpha \in R$ with $(\alpha )=\U0001d513$ and ${(\alpha )}^{2}=(2)$. Thus such a number $\alpha $ would have norm $2$, but the norm of $n+m\sqrt{-5}$ is ${n}^{2}+5{m}^{2}$ so it is clear that there are no algebraic integers^{} of norm $2$. Therefore $\U0001d513$ is not principal. Thus $R$ is not a PID.

Title | example of ring which is not a UFD |
---|---|

Canonical name | ExampleOfRingWhichIsNotAUFD |

Date of creation | 2013-03-22 15:08:19 |

Last modified on | 2013-03-22 15:08:19 |

Owner | alozano (2414) |

Last modified by | alozano (2414) |

Numerical id | 7 |

Author | alozano (2414) |

Entry type | Example |

Classification | msc 13G05 |

Synonym | example of a ring of integers which is not a UFD |

Related topic | DeterminingTheContinuationsOfExponent |

Defines | example of a number ring which is not a UFD |