By Krull's theorem for valuation rings, we may take $n>1$ and $G=\mathbb Z^n$ with lexicographic order to form a totally ordered Abelian group, and the resulting valuation domain $R$ is the ring. In this construction, is is possible to make choices to ensure the resulting ring is countable.
Notes: The classical Krull dimension of $R$ corresponds to the rank of $G$, in this case $n>1$.
| Name | Measure | |
|---|---|---|
| cardinality | $\aleph_0$ | |
| composition length | left: $\infty$ | right: $\infty$ |
| Krull dimension (classical) | $n$ | |
| uniform dimension | left: 1 | right: 1 |
| Name | Description |
|---|---|
| Idempotents | $\{0,1\}$ |
| Left singular ideal | $\{0\}$ |
| Left socle | $\{0\}$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |
| Right socle | $\{0\}$ |
| Zero divisors | $\{0\}$ |