A Singular Introduction to Commutative Algebra
Authors: Greuel, G.M., Pfister, Gerhard
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 About this Textbook

This book can be understood as a model for teaching commutative algebra, taking into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, it is shown how to handle it by computer. The computations are exemplified with the computer algebra system Singular, developed by the authors. Singular is a special system for polynomial computation with many features for global as well as for local commutative algebra and algebraic geometry. The text starts with the theory of rings and modules and standard bases with emphasis on local rings and localization. It is followed by the central concepts of commutative algebra such as integral closure, dimension theory, primary decomposition, Hilbert function, completion, flatness and homological algebra. There is a substantial appendix about algebraic geometry in order to explain how commutative algebra and computer algebra can be used for a better understanding of geometric problems. The book includes a CD with a distribution of Singular for various platforms (Unix/Linux, Windows, Macintosh), including all examples and procedures explained in the book. The book can be used for courses, seminars and as a basis for studying research papers in commutative algebra, computer algebra and algebraic geometry.
 Reviews

"…It is certainly no exaggeration to say that Greuel and Pfister's A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra, in which computational methods and results become central to how the subject is taught and learned. […] Among the great strengths and most distinctive features of Greuel and Pfister's book is a new, completely unified treatment of the global and local theories. The realization that the two cases could be combined to this extent was decisive in the design of the Singular system, making it one of the most flexible and most efficient systems of its type. The authors present the first systematic development of this unified approach in a textbook here, and this aspect alone is almost worth the price of admission. Another distinctive feature of this book is the degree of integration of explicit computational examples into the flow of the text. Strictly mathematical components of the development (often quite terse and written in a formal "theoremproof" style) are interspersed with parallel discussions of features of Singular and numerous Singular examples giving input commands, some extended programs in the Singular language, and output. […] Yet another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic. A synopsis of the table of contents will make this clear. […] Greuel and Pfister have written a distinctive an highly useful book that should be in the library of every commutative algebrais and algebraic geometer, expert and novice alike. I hope that it achieves the educational impact it deserves."
John B. Little, Monthly of The Mathematical Association of America, March 2004
"... The authors' most important new focus is the presentation of nonwell orderings that allow them the computational approach for local commutative algebra. The accompanying CDROM also contains all the examples of the book. ...
In fact the book provides an introduction to commutative algebra from a computational point of view. So it might be helpful for students and other interested readers (familiar with computers) to explore the beauties and difficulties of commutative algebra by computational experiences. In this respect the book is the one of the first samples of a new kind of textbooks in algebra."
P.Schenzel, Zentralblatt für Mathematik 1023.13001, 2003
"It is certainly no exaggeration to say that … A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra … . Among the great strengths and most distinctive features … is a new, completely unified treatment of the global and local theories. … Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike."
John B. Little, MAA, March 2004
"The aim of the book is … an introduction to commutative algebra with a view towards to algorithmic aspects and computational practice. … The authors’ most important new focus is the presentation of nonwell orderings that allow them the computational approach for local commutative algebra. … It might be helpful for students and other interested readers … to explore the beauties and difficulties of commutative algebra … . The book is one of the first samples of a new kind of textbooks in algebra."
Peter Schenzel, Zentralblatt MATH, Vol. 1023, 2003
 Table of contents (7 chapters)


Rings, Ideals and Standard Bases
Pages 188

Modules
Pages 89190

Noether Normalization and Applications
Pages 191237

Primary Decomposition and Related Topics
Pages 239274

Hilbert Function and Dimension
Pages 275312

Table of contents (7 chapters)
Bibliographic Information
 Bibliographic Information

 Book Title
 A Singular Introduction to Commutative Algebra
 Authors

 G.M. Greuel
 Gerhard Pfister
 Copyright
 2002
 Publisher
 SpringerVerlag Berlin Heidelberg
 Copyright Holder
 SpringerVerlag Berlin Heidelberg
 eBook ISBN
 9783662049631
 DOI
 10.1007/9783662049631
 Edition Number
 1
 Number of Pages
 XVII, 588
 Number of Illustrations
 44 b/w illustrations
 Topics