Indian Institute of Technology, Bombay

Powai, Mumbai 400076 India

1 | Rings and Modules | 3 |
---|---|---|

1.1 Ideals and Radicals | 3 | |

1.2 Polynomial rings and Localization of rings | 8 | |

1.3 Modules | 11 | |

1.4 Zariski Tolpology | 12 | |

Exercises | 14 | |

2 | Noetherian Rings | 17 |

2.1 Noetherian Rings and Modules | 17 | |

2.2 Primary Decomposition of Ideals | 19 | |

2.3 Artinian Rings and Modules | 23 | |

2.4 Krull's Principal Ideal Theorem | 27 | |

Exercises | 14 | |

3 | Integral Extensions | 32 |

3.1 Integral Extensions | 32 | |

3.2 Noether Normalization | 35 | |

3.3 Finiteness of Integral Closure | 38 | |

Exercises | 42 | |

4 | Dedekind Domains | 44 |

4.1 Dedekind Domains | 45 | |

4.2 Extensions of Primes | 50 | |

Exercises | 42 | |

A | Appendix: Primary Decomposition of Modules | 55 |

A.1 Associated Primes of Modules | 55 | |

A.2 Primary Decomposition of Modules | 58 | |

Exercises | 62 | |

References | 63 | |

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