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**The algebraic theory of modular systems. With a new introduction by Paul Roberts.
Reprint of the 1916 orig.**
*(English)*
Zbl 0802.13001

Cambridge Mathematical Library. Cambridge: Cambridge University Press. xxxi, 112 p. (1994).

This book is a welcome reissue of a classic in commutative algebra. Although the book is concerned mainly with an investigation of properties of ideals (called “modules” by the author) in a polynomial ring in the context of finding solutions of polynomial equations, it contains the origins of several other concepts which are today fundamental in commutative algebra. It is also a rich source of concrete examples, which are still relevant, since they supplement and provide the necessary counterweight to the modern abstract theory. Contrasted with the abstract, existential proofs found in commutative algebra today, the proofs in this text are constructive and concrete. One reason for the difference, of course, is that the modern theory deals with ideals in an abstract ring, while Macaulay dealt only with ideals in polynomial rings. However, even in the modern context, constructive proofs, particularly those amenable to computer methods, have acquired greater relevance because of the current interest in computational algebra and geometry.

As for describing the contents of the book in some detail, the reviewer can do no better than refer to the excellent introduction to the reissue written by Paul Roberts which includes a glossary translating some of the author’s terms into the modern language.

As for describing the contents of the book in some detail, the reviewer can do no better than refer to the excellent introduction to the reissue written by Paul Roberts which includes a glossary translating some of the author’s terms into the modern language.

Reviewer: Balwant Singh (Mumbai)

### MSC:

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

13-03 | History of commutative algebra |

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

13A15 | Ideals and multiplicative ideal theory in commutative rings |

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

13F20 | Polynomial rings and ideals; rings of integer-valued polynomials |

13C14 | Cohen-Macaulay modules |