A Singular Introduction to Commutative Algebra

1. Front Matter

   Pages I-XVII

   PDF

2. Rings, Ideals and Standard Bases

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 1-88

3. Modules

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 89-190

4. Noether Normalization and Applications

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 191-237

5. Primary Decomposition and Related Topics

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 239-274

6. Hilbert Function and Dimension

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 275-312

7. Complete Local Rings

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 313-334

8. Homological Algebra

   • Gert-Martin Greuel, Gerhard Pfister

   Pages 335-399

9. Back Matter

   Pages 401-588

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In theory there is no difference between theory and practice. In practice there is. Yogi Berra A SINGULAR Introduction to Commutative Algebra offers a rigorous intro­ duction to commutative algebra and, at the same time, provides algorithms and computational practice. In this book, we do not separate the theoretical and the computational part. Coincidentally, as new concepts are introduced, it is consequently shown, by means of concrete examples and general proce­ dures, how these concepts are handled by a computer. We believe that this combination of theory and practice will provide not only a fast way to enter a rather abstract field but also a better understanding of the theory, showing concurrently how the theory can be applied. We exemplify the computational part by using the computer algebra sys­ tem SINGULAR, a system for polynomial computations, which was developed in order to support mathematical research in commutative algebra, algebraic geometry and singularity theory. As the restriction to a specific system is necessary for such an exposition, the book should be useful also for users of other systems (such as Macaulay2 and CoCoA) with similar goals. Indeed, once the algorithms and the method of their application in one system is known, it is usually not difficult to transfer them to another system.

• algebra
• algebraic geometry
• algebraic varieties
• algorithms
• commutative algebra
• computer algebra
• geometry
• homomorphism

"…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 "theorem-proof" 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 non-well orderings that allow them thecomputational approach for local commutative algebra. The accompanying CD-ROM 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 non-well 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

 

• Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Germany

  Gert-Martin Greuel, Gerhard Pfister

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