*(English)*Zbl 0848.13001

Summary: The first edition of this standard textbook appeared 1965 (Zbl 0193.34701). – The author writes in the foreword of the second edition (1984): “In this second edition, I have added several topics, having mostly to do with commutative algebra and homological algebra, for instance: projective and injective modules, leading to an extended treatment of homological algebra, with derived functors, the Hilbert syzygy theorem, and a more thorough discussion of $K$-groups and Euler characteristics; the Quillen-Suslin theorem (previously Serre’s conjecture) that finite projective modules over a polynomial ring are free; the Weierstrass preparation theorem; the Hilbert polynomial in connection with filtered and graded modules; more material on tensor products, like flat modules and derivations; etc... – I have added a number of new exercises, especially in the chapters which are most likely to be of fundamental use, like the chapter on group theory, Noetherian rings, Galois theory, and tensor products.” In the foreword of the present third edition the author adds: “After almost a decade since the second edition. I find that the basic topics of algebra have become stable, with one exception. I have added two sections on elimination theory, complementing the existing section on the resultant. Algebraic geometry having progressed in many ways, it is now sometimes returning to older and harder problems, such as searching for the effective construction of polynomials vanishing on certain algebraic sets, and the older elimination procedures of last century serve as an introduction to those problems.

Except for this addition, the main topics of the book are unchanged from the second edition, but I have tried to improve the book in several ways.

First, some topics have been reordered. I was informed by readers and reviewers to the tension existing between having a textbook usable for relatively inexperienced students, and a reference book where results could easily be found in a systematic arrangement. I have tried to reduce this tension by moving all the homological algebra to a fourth part, and by integrating the commutative algebra with the chapter on algebraic sets and elimination theory, thus giving an introduction to different points of view leading toward algebraic geometry”.

##### MSC:

13Axx | General commutative ring theory |

13-01 | Textbooks (commutative algebra) |

14A05 | Relevant commutative algebra |

13Dxx | Homological methods (commutative rings) |

12-01 | Textbooks (field theory) |

15-01 | Textbooks (linear algebra) |

16-01 | Textbooks (associative rings and algebras) |

18-01 | Textbooks (category theory) |

20-01 | Textbooks (group theory) |

00A05 | General mathematics |