Let $\{I_i\mid i\in \mathbb N\}$ be a partition of $\mathbb N$ such that $|I_i|<|I_{i+1}|$. From $\mathbb Q[x_0, x_1,\ldots]$, consider the prime ideals $P_i=(\{x_j\mid j\in I_i\})$. The ring $R$ is the semilocalization at the intersection of the complements of the $P_i$'s.
Keywords localization polynomial ring
| Name | Measure | |
|---|---|---|
| cardinality | $\aleph_0$ | |
| global dimension | left: $\infty$ | right: $\infty$ |
| Krull dimension (classical) | $\infty$ |
| Name | Description |
|---|---|
| Idempotents | $\{0,1\}$ |
| Left singular ideal | $\{0\}$ |
| Left socle | $\{0\}$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |
| Right socle | $\{0\}$ |
| Zero divisors | $\{0\}$ |