##
**Lessons on rings, modules and multiplicities.
Reprint of the 1968 hardback ed.**
*(English)*
Zbl 1185.13001

Cambridge: Cambridge University Press (ISBN 978-0-521-09807-6/pbk). xiv, 444 p. (2008).

In his long academic career as active researcher and devoted University teacher, the renowned British algebraist and geometer Douglas G. Northcott (1916–2005) has published seven outstanding textbooks, all of which became both popular standard references and venerable classics. Northcott’s textbooks, which still captivate any reader by their expository mastery, are mainly of introductory nature, thereby providing easily readable, systematic and topical accounts of the fundamentals of commutative algebra, homological algebra, and local algebraic geometry, respectively. Indeed, D. G. Northcott has always set high value on introducing students and young researchers to relatively recent and advanced topics in modern algebra and geometry, with the aim of providing a bridge between undergraduate and postgraduate study, on the one hand, and between classical and contemporary research topics on the other.

After his successful introductory texts on “Ideal theory” [Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge: At the University Press VIII, 111 p. (1953; Zbl 0052.26801)] and on homological algebra [“An Introduction to Homological Algebra” Cambridge: University Press 1960, XI, 282 p. (1960; Zbl 0116.01401)], respectively, Northcott’s “Lessons on Rings, Modules and Multiplicities” was the third book in the row of his seven expository masterpieces. First published in 1968, this textbook had grown out of the author’s lectures and seminars held at the University of Sheffield, UK, in the 1960s. As the title indicates, this book provides a very special introduction to selected topics in modern commutative algebra. Moreover, the particular title “Lessons on Rings, Modules and Multiplicities” was chosen partly because a strong emphasis has been placed on the instructional aspects, meaning that the author has endeavoured to present the chosen material in a manner which is to make it easy to assimilate.

The book under review is a digitally printed version of the original edition (Zbl 0159.33001) and, after forty years, the first reprint of this classic at all. Although we may therefore refer to the review of the original from 1968, a few retrospective comments on Northcott’s classic text, and on the current reprint, might be quite appropriate, just in order to characterize the place of this particular textbook within the contemporary literature on the subject. Let us first briefly recall the contents of the nine chapters of the book, which are as follows:

1. Introduction to the basic ideas. – This chapter provides the fundamentals on rings and modules in a general setting, that is, largely without assuming commutative rings.

2. Prime ideals and primary submodules. – This chapter also treats integral ring extensions, primary decompositions for modules, graded rings and modules, and homogeneous primary decompositions.

3. Rings and modules of fractions. – The objects of study are here localization functors and their properties.

4. Noetherian rings and modules. – This chapter deals with Noetherian and Artinian rings and modules, respectively, including the graded case, length functions, the Artin-Rees Lemma, the notion of Krull dimension, and the fundamental properties of local and semi-local rings.

5. The theory of grade. – This chapter introduces the grade of a module and discusses its basic properties, mainly for special classes of ground rings.

6. Hilbert rings and the Theorem of Zeros. – Here Hilbert’s Nullstellensatz and its ring-theoretic generalization are the main objects of study.

7. Multiplicity theory. – This chapter is devoted to the development of a general theory of algebraic multiplicities, largely presented in the context of non-commutative ground rings. The reader meets here multiplicity systems, the multiplicity symbol, Lech’s limit formula, Hilbert functions. Samuel’s limit formula, and further properties of multiplicity-functions.

8. The Koszul complex. – After some generalities on chain complexes of modules, the construction of the Koszul complex, its basic properties, and its use in both multiplicity theory and theory of grade are thoroughly explained.

9. Filtered rings and modules. – This final chapter discusses the topological aspects of rings and modules, with a special view toward ideal-adic filtrations, topologies, and completions.

Each chapter ends with a number of related exercises providing supplements and illustrating concrete examples.

As one can see from the table of contents, Northcott’s textbook has maintained its topicality over the past forty years. It offers a unique, tailor-made blend of topics from the more advanced standard texts of N. Bourbaki, M. Nagata, P. Samuel and O. Zariski, and H. Matsumura, enriched by additional material from the works of Auslander-Buchsbaum (concerning multiplicity theory and the use of the Koszul complex) and of Goldman-Krull (with respect to the theory of Hilbert rings). In this regard, Northcott’s introductory textbook may not only serve as an excellent first reading, and as a perfect companion to the more encyclopedic standard treatises in the field, both old and new, but also as a friendly mediator between those, especially for students and non-specialists. Another feature of Northcott’s individual approach, distinguishing it from most other expositions of commutative and local algebra, is that it offers the theory in the greatest possible generality, and that nearly effortless and in a highly systematizing manner.

With a view to these outstanding features, Northcott’s classic on rings, modules and multiplicities was, is, and will remain one of the most useful primers in modern abstract algebra. New generations of students and teachers will certainly appreciate the gratifying circumstance that this masterful textbook has been made available again, this time in affordable paperback form.

After his successful introductory texts on “Ideal theory” [Cambridge Tracts in Mathematics and Mathematical Physics. Cambridge: At the University Press VIII, 111 p. (1953; Zbl 0052.26801)] and on homological algebra [“An Introduction to Homological Algebra” Cambridge: University Press 1960, XI, 282 p. (1960; Zbl 0116.01401)], respectively, Northcott’s “Lessons on Rings, Modules and Multiplicities” was the third book in the row of his seven expository masterpieces. First published in 1968, this textbook had grown out of the author’s lectures and seminars held at the University of Sheffield, UK, in the 1960s. As the title indicates, this book provides a very special introduction to selected topics in modern commutative algebra. Moreover, the particular title “Lessons on Rings, Modules and Multiplicities” was chosen partly because a strong emphasis has been placed on the instructional aspects, meaning that the author has endeavoured to present the chosen material in a manner which is to make it easy to assimilate.

The book under review is a digitally printed version of the original edition (Zbl 0159.33001) and, after forty years, the first reprint of this classic at all. Although we may therefore refer to the review of the original from 1968, a few retrospective comments on Northcott’s classic text, and on the current reprint, might be quite appropriate, just in order to characterize the place of this particular textbook within the contemporary literature on the subject. Let us first briefly recall the contents of the nine chapters of the book, which are as follows:

1. Introduction to the basic ideas. – This chapter provides the fundamentals on rings and modules in a general setting, that is, largely without assuming commutative rings.

2. Prime ideals and primary submodules. – This chapter also treats integral ring extensions, primary decompositions for modules, graded rings and modules, and homogeneous primary decompositions.

3. Rings and modules of fractions. – The objects of study are here localization functors and their properties.

4. Noetherian rings and modules. – This chapter deals with Noetherian and Artinian rings and modules, respectively, including the graded case, length functions, the Artin-Rees Lemma, the notion of Krull dimension, and the fundamental properties of local and semi-local rings.

5. The theory of grade. – This chapter introduces the grade of a module and discusses its basic properties, mainly for special classes of ground rings.

6. Hilbert rings and the Theorem of Zeros. – Here Hilbert’s Nullstellensatz and its ring-theoretic generalization are the main objects of study.

7. Multiplicity theory. – This chapter is devoted to the development of a general theory of algebraic multiplicities, largely presented in the context of non-commutative ground rings. The reader meets here multiplicity systems, the multiplicity symbol, Lech’s limit formula, Hilbert functions. Samuel’s limit formula, and further properties of multiplicity-functions.

8. The Koszul complex. – After some generalities on chain complexes of modules, the construction of the Koszul complex, its basic properties, and its use in both multiplicity theory and theory of grade are thoroughly explained.

9. Filtered rings and modules. – This final chapter discusses the topological aspects of rings and modules, with a special view toward ideal-adic filtrations, topologies, and completions.

Each chapter ends with a number of related exercises providing supplements and illustrating concrete examples.

As one can see from the table of contents, Northcott’s textbook has maintained its topicality over the past forty years. It offers a unique, tailor-made blend of topics from the more advanced standard texts of N. Bourbaki, M. Nagata, P. Samuel and O. Zariski, and H. Matsumura, enriched by additional material from the works of Auslander-Buchsbaum (concerning multiplicity theory and the use of the Koszul complex) and of Goldman-Krull (with respect to the theory of Hilbert rings). In this regard, Northcott’s introductory textbook may not only serve as an excellent first reading, and as a perfect companion to the more encyclopedic standard treatises in the field, both old and new, but also as a friendly mediator between those, especially for students and non-specialists. Another feature of Northcott’s individual approach, distinguishing it from most other expositions of commutative and local algebra, is that it offers the theory in the greatest possible generality, and that nearly effortless and in a highly systematizing manner.

With a view to these outstanding features, Northcott’s classic on rings, modules and multiplicities was, is, and will remain one of the most useful primers in modern abstract algebra. New generations of students and teachers will certainly appreciate the gratifying circumstance that this masterful textbook has been made available again, this time in affordable paperback form.

Reviewer: Werner Kleinert (Berlin)

### MSC:

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

01A75 | Collected or selected works; reprintings or translations of classics |

13Axx | General commutative ring theory |

13Cxx | Theory of modules and ideals in commutative rings |

13Hxx | Local rings and semilocal rings |

13Bxx | Commutative ring extensions and related topics |

13C05 | Structure, classification theorems for modules and ideals in commutative rings |

13Exx | Chain conditions, finiteness conditions in commutative ring theory |

13H15 | Multiplicity theory and related topics |

13J10 | Complete rings, completion |