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.

This book can be understood as a model for teaching commutative algebra, and takes into account modern developments such as algorithmic and computational aspects. As soon as a new concept is introduced, the authors show how the concept can be worked on using a 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 book includes a CD containing Singular as well as the examples and procedures explained in the book.

1 Rings, Ideals and Standard Bases 1.1 Rings, Polynomials and Ring Maps 1.2 Monomial Orderings 1.3 Ideals and Quotient Rings 1.4 Local Rings and Localization 1.5 Rings Associated to Monomial Orderings 1.6 Normal Forms and Standard Bases 1.7 The Standard Basis Algorithm 1.8 Operations on Ideals and their Computation 1.8.1 Ideal membership 1.8.2 Intersection with subrings (elimination of variables) 1.8.3 Zariski closure of the image 1.8.4 Solvability of polynomial equations 1.8.5 Solving polynomial equations 1.8.6 Radical membership 1.8.7 Intersection of ideals 1.8.8 Quotient of ideals 1.8.9 Saturation 1.8.10 Kernel of a ring map 1.8.11 Algebraic dependence and subalgebra membership 2. Modules 2.1 Modules, Submodules and Homomorphisms 2.2 Graded Rings and Modules 2.3 Standard Bases for Modules 2.4 Exact Sequences and free Resolutions 2.5 Computing Resolutions and the Syzygy Theorem 2.6 Modules over Principal Ideal Domains 2.7 Tensor Product 2.8 Operations with modules 2.8.1 Module membership problem 2.8.2 Elimination of module components 2.8.3 Quotients of submodules 2.8.4 Kernel of a module homomorphism 3. Noether Normalization and Applications 3.1 Finite and Integral Extensions 3.2 The Integral Closure 3.3 Dimension 3.4 Noether Normalization 3.5 Applications 3.6 An Algorithm to Compute the Normalization 3.7 Procedures 4. Primary Decomposition and Related Topics 4.1 The Theory of Primary Decomposition 4.2 Zero--dimensional Primary Decomposition 4.3 Higher Dimensional Primary Decomposition 4.4 The Equidimensional Part of an Ideal 4.5 The Radical 4.6 Procedures 5. Hilbert Function 5.1 The Hilbert Function and the Hilbert Polynomial 5.2 Examples and Computation of the Hilbert--Poincare Series 5.3 Properties of the Hilbert Polynomial 5.4 Filtrations and the Lemma of Artin--Rees 5.5 The Hilbert--Samuel Function 5.6 Characterization of the Dimension of Local Rings 5.7 Singular Locus 6. Complete Local Rings 6.1 Formal Power Series Rings 6.2 Weierstrass Preparation Theorem 6.3 Completions 6.4 Standard bases 7. Homological Algebra 7.1 Tor 7.2 Fitting Ideals 7.3 Flatness 7.4 Local Criteria for Flatness 7.5 Flatness and Standard Bases 7.6 Koszul Complex 7.7 Cohen-Macaulay Rings 7.8 Further Characterization of Cohen-Macaulayness A. Geometric Background A.1 Introduction by Pictures A.2 Affine algebraic varieties A.3 Spectrum and Affine Schemes A.4 Projective Varieties and Projective Schemes A.5 Morphisms between Varieties A.6 Projective Morphisms and elimination A.7 Local versus Global Properties A.8 Singularities B. SINGULAR - A Short Introduction B.1 Downloading Instructions B.2 Getting Started B.3 Procedures and Libraries B.4 Data Types B.5 Functions B.6 Control Structures B.7 System variables SINGULAR Reference Manual Index Glossary