By Krull's theorem for valuation rings, we may take $n>1$ and $G=\mathbb Z^n$ with lexicographic order to form a totally ordered Abelian group, and the resulting valuation domain $R$ is the ring. In this construction, is is possible to make choices to ensure the resulting ring is countable.

Notes: The classical Krull dimension of $R$ corresponds to the rank of $G$, in this case $n>1$.

Known Properties

Legend

- = has the property
- = does not have the property
- = information not in database

Name | Measure | |
---|---|---|

cardinality | $\aleph_0$ | |

composition length | left: $\infty$ | right: $\infty$ |

Krull dimension (classical) | $n$ | |

uniform dimension | left: 1 | right: 1 |

Name | Description |
---|---|

Idempotents | $\{0,1\}$ |

Left singular ideal | $\{0\}$ |

Left socle | $\{0\}$ |

Nilpotents | $\{0\}$ |

Right singular ideal | $\{0\}$ |

Right socle | $\{0\}$ |

Zero divisors | $\{0\}$ |