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 book includes a CD with a version of Singular for various platforms (Unix/Linux, Windows, Macintosh),…mehr
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 book includes a CD with a version 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.
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Inhaltsangabe
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 Pr.
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 Pr.
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