## Undergraduate Commutative AlgebraCommutative algebra is at the crossroads of algebra, number theory and algebraic geometry. This textbook is affordable and clearly illustrated, and is intended for advanced undergraduate or beginning graduate students with some previous experience of rings and fields. Alongside standard algebraic notions such as generators of modules and the ascending chain condition, the book develops in detail the geometric view of a commutative ring as the ring of functions on a space. The starting point is the Nullstellensatz, which provides a close link between the geometry of a variety V and the algebra of its coordinate ring A=k[V]; however, many of the geometric ideas arising from varieties apply also to fairly general rings. The final chapter relates the material of the book to more advanced topics in commutative algebra and algebraic geometry. It includes an account of some famous 'pathological' examples of Akizuki and Nagata, and a brief but thought-provoking essay on the changing position of abstract algebra in today's world. |

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### Contents

g Discussion of contents | 12 |

Basics | 19 |

g Plenty of prime ideals | 27 |

g Exact sequences | 44 |

Finite extensions and Noether normalisation | 58 |

The Nullstellensatz and Spec A | 70 |

Exercises to Chapter 5 | 82 |

Exercises to Chapter 6 | 92 |

Exercises to Chapter 7 | 110 |

Exercises to Chapter 8 | 126 |

Akizukis example | 139 |

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### Common terms and phrases

abstract algebra algebraic geometry algebraically closed field ascending chain axiom closed set codimension coefficients commutative algebra condition consider contains coordinate ring Corollary counterexample defined definition discussed easy exact sequence example Exercises to Chapter exists extension ring factorisation factors field extension field of fractions finite A-module finite set follows Galois gives Hint implies integral closure integral dependence relation integral domain intersection isomorphism linear localisation maximal element maximal ideal minimal nonzero prime module monic multiplicative set nilpotent Noether normalisation Noetherian ring nonsingular normal Nullstellensatz number field obviously polynomial ring primary decomposition prime element prime number Proof Proposition Let prove rational functions residue field ring homomorphism ring of functions S-lA satisfies Spec submodules subring subset subvarieties Supp Suppose surjective unique valuation ring vector space write X C kn Zariski topology zerodivisors Zorn's lemma