$R$ is the restricted direct product of $\mathbb R$ and the $p$-adic rings $\mathbb Q_p$. That is, it is the subring of $\mathbb R\times\prod_p \mathbb Q_p$ of elements for which all but finitely many coordinates lie in $\mathbb Z_p$ for each respective position indexed by $p$.
Keywords direct product subring
| Name | Measure | |
|---|---|---|
| Krull dimension (classical) | $\infty$ |
| Name | Description |
|---|---|
| Jacobson radical | $\{0\}$ |
| Left singular ideal | $\{0\}$ |
| Left socle | $\mathbb R\oplus (\bigoplus \mathbb Q_p)$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |
| Right socle | $\mathbb R\oplus (\bigoplus \mathbb Q_p)$ |