The quotient ring $\mathbb Q[x,y]/(x^2, xy)$

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

- H. C. Hutchins. Examples of commutative rings. (1981) @ Example 24 p 60
- I. Kaplansky. Commutative rings. (1974) @ Exercise 6 p 102

Known Properties

Legend

- = has the property
- = does not have the property
- = information not in database

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 | $(x)$ |