Let $X$ be a zero-dimensional topological space such that the Stone-Čech compactification $\beta X$ is not zero-dimensional. The required ring is $R=C(X)$, the set of continuous functions from $X$ into $\mathbb R$. (A topological space is called zero-dimensional if it has a base of clopen sets.)
Keywords ring of functions
| Name | Measure | |
|---|---|---|
| composition length | left: $\infty$ | right: $\infty$ |
| Name | Description |
|---|---|
| Jacobson radical | $\{0\}$ |
| Left singular ideal | $\{0\}$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |