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**Steps in commutative algebra.**
*(English)*
Zbl 0703.13001

London Mathematical Society Student Texts, 19. Cambridge etc.: Cambridge University Press. xi, 321 p. £30.00/hbk; $ 49.50/hbk; £10.95/pbk; $ 17.95/pbk (1990).

This textbook is intended as a gentle introduction to more advanced texts in commutative algebra; it is aimed at graduate algebraists and has grown out of lectures given as part of an MSc course at Sheffield University. The author is to be congratulated on producing a readable, error-free book.

The first half of the book (nine chapters) contains expected topics like rings and modules of fractions, Hilbert’s basis theorem, primary decomposition for ideals of a Noetherian ring and submodules of a Noetherian module. Notions from homological algebra like projective modules and Hom are just included, but readers are encouraged to look elsewhere for \(\otimes\), flatness and inverse limits. Graded rings and modules are not studied so Krull’s intersection theorem has the classical proof which does not use the lemma of Artin-Rees. - The next two chapters include the structure theorem for finitely generated modules over a PID as an application of the earlier theory and the use of this to establish canonical forms for square matrices over a field. - The author chooses dimension theory as the subject of the final four chapters and a bias towards the relevance of commutative algebra to algebraic geometry. - So, after an introduction to field theory which includes transcendence degree and degree of an algebraic extension, he moves in chapter 13 to integral extensions of rings (unusually this is prefaced by a short digression into determinants of square matrices over a commutative ring); here he proves the lying-over, going-up and going-down theorems. Next affine algebras over a field are studied centred on Hilbert’s Nullstellensatz and Noether’s normalization theorem. - The final chapter studies dimension of Noetherian rings in general, proves Krull’s generalized principal ideal theorem, introduces systems of parameters for a local ring and establishes Hilbert’s syzygy theorem for a regular local ring. As in the first half of the book the student is referred to other specified texts for further reading in such topics as Galois theory, exterior algebras, valuations, Hilbert-Samuel polynomials, R-sequences and the homological characterization of regular local rings.

The first half of the book (nine chapters) contains expected topics like rings and modules of fractions, Hilbert’s basis theorem, primary decomposition for ideals of a Noetherian ring and submodules of a Noetherian module. Notions from homological algebra like projective modules and Hom are just included, but readers are encouraged to look elsewhere for \(\otimes\), flatness and inverse limits. Graded rings and modules are not studied so Krull’s intersection theorem has the classical proof which does not use the lemma of Artin-Rees. - The next two chapters include the structure theorem for finitely generated modules over a PID as an application of the earlier theory and the use of this to establish canonical forms for square matrices over a field. - The author chooses dimension theory as the subject of the final four chapters and a bias towards the relevance of commutative algebra to algebraic geometry. - So, after an introduction to field theory which includes transcendence degree and degree of an algebraic extension, he moves in chapter 13 to integral extensions of rings (unusually this is prefaced by a short digression into determinants of square matrices over a commutative ring); here he proves the lying-over, going-up and going-down theorems. Next affine algebras over a field are studied centred on Hilbert’s Nullstellensatz and Noether’s normalization theorem. - The final chapter studies dimension of Noetherian rings in general, proves Krull’s generalized principal ideal theorem, introduces systems of parameters for a local ring and establishes Hilbert’s syzygy theorem for a regular local ring. As in the first half of the book the student is referred to other specified texts for further reading in such topics as Galois theory, exterior algebras, valuations, Hilbert-Samuel polynomials, R-sequences and the homological characterization of regular local rings.

Reviewer: D.Kirby

### MSC:

13-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra |

13-02 | Research exposition (monographs, survey articles) pertaining to commutative algebra |

12-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory |

14-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry |