$k[x,y]/(x^2, xy)$ where $k$ is a countably infinite field.
Notes: The zero ideal is not primary, but it satisfies the false naive definition of primary as being $xy\in I$ implies $x^n\in I$ or $y^n\in I$.
Keywords polynomial ring quotient ring
| Name | Measure | |
|---|---|---|
| cardinality | $\aleph_0$ | |
| composition length | left: $\infty$ | right: $\infty$ |
| Krull dimension (classical) | 1 |
| Name | Description |
|---|---|
| Idempotents | $\{0,1\}$ |
| Left socle | $(x)$ |
| Right socle | $(x)$ |