For a sequence of non-negative integers $\{n_i\}$ such that $n_0 = 0$ and $n_r \ge 2 n_{r-1} + 2$, take a transcendental element $z_0 \in \Bbb Z_2$ of form $z_0 = \sum\limits_{r = 0}^\infty a_r 2^{n_r}$, where $a_r \in \{1, \ldots, 2-1\}$. Set $z_{r + 1} = (z_r - a_r) / 2^{n_r}$. The ring is $\Bbb Z[2 (z_0 - a_0), \{(z_i - a_i)^2 | i = 0, 1, \ldots\}] \subset \Bbb Z_2$.

Notes: Details in https://arxiv.org/pdf/alg-geom/9503017.pdf

Keywords power series ring

- M. Reid. Akizuki's counterexample. (1995) @ (whole paper)

Known Properties

Legend

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

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

Krull dimension (classical) | 1 |

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

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

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

Left socle | $\{0\}$ |

Nilpotents | $\{0\}$ |

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

Right socle | $\{0\}$ |

Zero divisors | $\{0\}$ |