Take $S=R_{72}$ and the idempotent $e=E_{11}+E_{33}+E_{44}+E_{55}$. This turns out to be a full idempotent, and the required ring is the corner ring $R=eSe$. In other words, $R=\begin{bmatrix}a&p&0&0 \\ 0&r&0&0 \\ 0&0&r&s \\ 0&0&0&a\end{bmatrix}$ for $a,p,r,s\in F_2$.
Keywords basic ring matrix ring subring
| Name | Measure | |
|---|---|---|
| cardinality | 16 | |
| Krull dimension (classical) | 0 |
| Name | Description |
|---|---|
| Idempotents | All elements with $a=r=1$. |
| Jacobson radical | Subset of strictly upper triangular matrices |
| Left singular ideal | Subset of strictly upper triangular matrices |
| Left socle | Subset of strictly upper triangular matrices |
| Nilpotents | Subset of strictly upper triangular matrices |
| Right singular ideal | Subset of strictly upper triangular matrices |
| Right socle | Subset of strictly upper triangular matrices |