Let $F$ be the field of two elements, and consider a countably infinite direct sum of copies of $F$. This is a countable boolean ring (without unity). The required ring is the subring generated by this subrng of $\prod F$ and the identity of $\prod F$. This is still countable. Alternatively, the ring of "eventually constant" sequences in $\prod F$.
Keywords direct product subring
| Name | Measure | |
|---|---|---|
| cardinality | $\aleph_0$ | |
| composition length | left: $\infty$ | right: $\infty$ |
| global dimension | left: 1 | right: 1 |
| Krull dimension (classical) | 0 | |
| weak global dimension | 0 |
| Name | Description |
|---|---|
| Jacobson radical | $\{0\}$ |
| Left singular ideal | $\{0\}$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |