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**A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann.
2nd extended ed.**
*(English)*
Zbl 1133.13001

Berlin: Springer (ISBN 978-3-540-73541-0/hbk). xx, 689 p. with CD-ROM. (2007).

The book under review is the second, extended edition of the first printing, see [A Singular introduction to commutative algebra. With contributions by Olaf Bachmann, Christoph Lossen and Hans Schönemann. (Berlin): Springer. (2002; Zbl 1023.13001)], published in 2002. During these five years the computeralgebra system Singular (developed with a strong input of the authors of the book) became one of the most popular system for computations in commutative algebra and algebraic geometry. The actual version 3.0.4 is freely available under the GNU public license for various platforms via http://www.singular.uni-kl.de. In particular, during the last five years the development of Singular was supported by an international community that has created a large number of proceedures for some special computations available as libraries in Singular. Moreover, there is a kernel extension providing Gröbner bases algorithms and implementations for ideals and modules in non-commutative algebras with a certain condition for non-commutativity. Most of this was contributed by V. Lewandowsky.

The second edition of the book reflects this rapid development of Singular in the following way: The Appendix C of the book, Singular - A short Introduction, concipated as a crash course of Singular’s programming language, is rewritten corresponding to the recent version. It covers more examples on how to write labraries and how to communicate with other systems, extending those of the first edition (Mathematica, Maple, MuPAD) by GAP and SAGE. A new CD is included containing all the examples of the book and most of the Singular-libaries. That means, the book does cover not only the theoretical background but also the actual version of the software with the examples in order to become familar with practical experiences. Moreover, the CD is completed with a lot of additional material, the manual (also available via the program), links to papers, other software resources and - of course - the sources and the binaries for Singular.

The major extensions in the text are the following: (1) There is a new section of Chapter 1 “Rings, Ideals and Standard Bases” about non-commutative Gröbner bases. (2) Chapter 4 “Primary Decomposition and Related Topics” is extended by two new sections about characteristic and triangular sets with the corresponding decomposition algorithm. Characteristic sets are useful in order to compute the minimal associated prime ideals of an ideal. Triangular sets are a basic tool for the symbolic pre-processing to solve zero-dimensional systems of polynomial equations. (3) There is a new Appendix B “Polynomial Factorization” concerning univariate factorization over \(\mathbb F_p\) and \(\mathbb Q\) and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of \(\mathbb Q.\)

In addition to what is said about the first edition the book is distinctive and highly useful in order to explore the beauties and difficulties of commutative algebra and algebraic geometry by computational and theoretical insights. It provides the theory in a clever way as well as all the requirements for practical experiments conceived for Singular but nevertheless there is no strict restriction in order to use the material with different computer algebra systems. It is highly recommended for all – students and researchers – who are interested in practical computations of their algebraic interests as well as for introductory research projects for students.

The second edition of the book reflects this rapid development of Singular in the following way: The Appendix C of the book, Singular - A short Introduction, concipated as a crash course of Singular’s programming language, is rewritten corresponding to the recent version. It covers more examples on how to write labraries and how to communicate with other systems, extending those of the first edition (Mathematica, Maple, MuPAD) by GAP and SAGE. A new CD is included containing all the examples of the book and most of the Singular-libaries. That means, the book does cover not only the theoretical background but also the actual version of the software with the examples in order to become familar with practical experiences. Moreover, the CD is completed with a lot of additional material, the manual (also available via the program), links to papers, other software resources and - of course - the sources and the binaries for Singular.

The major extensions in the text are the following: (1) There is a new section of Chapter 1 “Rings, Ideals and Standard Bases” about non-commutative Gröbner bases. (2) Chapter 4 “Primary Decomposition and Related Topics” is extended by two new sections about characteristic and triangular sets with the corresponding decomposition algorithm. Characteristic sets are useful in order to compute the minimal associated prime ideals of an ideal. Triangular sets are a basic tool for the symbolic pre-processing to solve zero-dimensional systems of polynomial equations. (3) There is a new Appendix B “Polynomial Factorization” concerning univariate factorization over \(\mathbb F_p\) and \(\mathbb Q\) and algebraic extensions, as well as multivariate factorization over these fields and over the algebraic closure of \(\mathbb Q.\)

In addition to what is said about the first edition the book is distinctive and highly useful in order to explore the beauties and difficulties of commutative algebra and algebraic geometry by computational and theoretical insights. It provides the theory in a clever way as well as all the requirements for practical experiments conceived for Singular but nevertheless there is no strict restriction in order to use the material with different computer algebra systems. It is highly recommended for all – students and researchers – who are interested in practical computations of their algebraic interests as well as for introductory research projects for students.

Reviewer: Peter Schenzel (Halle)

### MSC:

13-04 | Software, source code, etc. for problems pertaining to commutative algebra |

13Pxx | Computational aspects and applications of commutative rings |

14-04 | Software, source code, etc. for problems pertaining to algebraic geometry |

13P10 | Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) |

14Qxx | Computational aspects in algebraic geometry |

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

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

68W30 | Symbolic computation and algebraic computation |

68-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to computer science |

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