$R=\prod_{i=1}^\infty \mathbb Z$, where addition is done pointwise, but the product $ab$ is done via the rule $(ab)_n := \sum \{\gcd(i,j)a_ib_j \mid \mathrm{lcm}(i,j)=n\}$
(Nothing was retrieved.)
| Name | Description |
|---|---|
| Idempotents | $\{0,1\}$ |
| Jacobson radical | $\{0\}$ |
| Left singular ideal | $\{0\}$ |
| Left socle | $\{0\}$ |
| Nilpotents | $\{0\}$ |
| Right singular ideal | $\{0\}$ |
| Right socle | $\{0\}$ |