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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.
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