The monoid ring $\Bbb Q[M]$ where $M$ is the quotient monoid $F(a,b,c,d,x,y,u,v)/С$, $F(\ldots)$ is the free monoid on those $8$ symbols and $C$ is the congruence generated by $ax = by$, $cx = dy$, $au = bv$. (See section 2 of Malcev's paper cited here.)

Notes: This is a domain that is not a subring of any division ring.

Keywords monoid ring

- A. Malcev. On the immersion of an algebraic ring into a field. (1937) @ Section 3

Symmetric properties

Asymmetric properties

Legend

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

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

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

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

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

Jacobson radical | $\{0\}$ |

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

Left socle | $\{0\}$ |

Nilpotents | $\{0\}$ |

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

Right socle | $\{0\}$ |

Zero divisors | $\{0\}$ |