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
| 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 | $\{0\}$ |