Let $S$ be the semigroup of positive pairs of rationals under addition, along with $(0,0)$, and $k$ be a field (for concreteness we suppose the field of two elements, but any field works.) The semigroup algebra $R=k[S]$ is the required ring.

Keywords semigroup ring

- P. M. Cohn. Bezout rings and their subrings. (1968) @ Example after Theorem 2.4 on p 256

Known Properties

Legend

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

(Nothing was retrieved.)

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

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

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

Left socle | $\{0\}$ |

Nilpotents | $\{0\}$ |

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

Right socle | $\{0\}$ |

Zero divisors | $\{0\}$ |