Steps in commutative algebra. 2nd ed. (English) Zbl 0969.13001

London Mathematical Society Student Texts. 51. Cambridge: Cambridge University Press. xii, 355 p. (2000).
This book is intended as an introduction to commutative algebra for students who have taken a basic algebra course but who are not expected to know about ideals, modules, categories, or homological algebra. The book aims to prepare such students for studying from standard commutative algebra texts, such as the book by H. Matsumura [“Commutative Algebra”, 2nd ed., Mathematics Lecture Note Series 56. Reading, Massachusetts, etc.: The Benjamin/Cummings Publishing company, Inc. (1980; Zbl 0441.13001)]. This is the second edition of the book under review, which includes two new chapters and minor corrections and additions to the first fifteen. To quote from D. Kirby’s review of the first edition of the book under review (Cambridge 1990; Zbl 0703.13001): “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. \(\dots\) 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 [next] four chapters \(\dots\) 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 \(\dots\) 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 [fifteenth] 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.”
The first of the new chapters is devoted to regular sequences and grade. The author defines the relevant concepts, proves the basic properties of depth and grade, and proves that for ideals grade is bounded by height and for modules depth is bounded by dimension.
The final chapter is about Cohen-Macaulay rings: These are defined as commutative Noetherian rings in which grade and height are equal for all ideals. The author proves the characterization of Cohen-Macaulay rings in terms of unmixedness, and in terms of grade equaling height for maximal ideals. And he proves that polynomial rings over Cohen-Macaulay rings are themselves Cohen-Macaulay. Finally, the bibliography (of commutative algebra textbooks) is updated for this edition as well.


13-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to commutative algebra
13-02 Research exposition (monographs, survey articles) pertaining to commutative algebra
13C15 Dimension theory, depth, related commutative rings (catenary, etc.)
12-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to field theory
14-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic geometry
13Cxx Theory of modules and ideals in commutative rings
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)