by** Gert-Martin Greuel **and** Gerhard Pfister **with contributions by** Olaf Bachmann**, **Christoph Lossen**, and** Hans Schönemann**

This book has been written to learn commutative algebra in a new style, 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, which was 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 book can be used for courses, seminars and as a basis for studying research papers in commutative algebra, computer 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. An appendix is devoted to algebraic geometry to show how geometric problems can be understood using commutative algebra. The book includes a CD with a distribution of Singular for various platforms (Unix/Linux, Windows, Macintosh) and which contains all examples and procedures explained in the text.

Note that this book is not an introduction to Singular, but to commutative algebra, with a view towards algebraic geometry and singularity theory, and how to use Singular for effective computations in this field. For documentation about Singular itself please have a look at the publications site.