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Release

February 2010: Release of SINGULAR version 3-1-1. More

Jenks Prize

July 2004: The Richard D. Jenks Prize for Excellence in Software Engineering for Computer Algebra was awarded to the Singular team. More

# Book: "A Singular Introduction to Commutative Algebra"

## Table of contents:

Preface (ps-file) |

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 |

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 on Modules and their Computation |

2.8.1 Module Membership Problem |

2.8.2 Intersection with Free Submodules |

2.8.3 Intersection of Submodules |

2.8.4 Quotients of Submodules |

2.8.5 Radical and Zerodivisors of Modules |

2.8.6 Annihilator and Support |

2.8.7 Kernel of a Module Homomorphism |

2.8.8 Solving Systems of Linear Equations |

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 and Dimension |

5.1 The Hilbert Function and the Hilbert Polynomial |

5.2 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 and Exactness |

7.2 Fitting Ideals |

7.3 Flatness |

7.4 Local Criteria for Flatness |

7.5 Flatness and Standard Bases |

7.6 Koszul Complex and Depth |

7.7 Cohen-Macaulay Rings |

7.8 Further Characterization of Cohen-Macaulayness |

A. Geometric Background |

A.1 Introduction by Pictures (ps-file) |

A.2 Affine Algebraic Varieties |

A.3 Spectrum and Affine Schemes |

A.4 Projective Varieties |

A.5 Projective Schemes and Varieties |

A.6 Morphisms between Varieties |

A.7 Projective Morphisms and Elimination |

A.8 Local versus Global Properties |

A.9 Singularities |

B. SINGULAR - A Short Introduction (ps-file) |

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 |

B.8 Libraries |

B.9 SINGULAR and Maple |

B.10 SINGULAR and Mathematica |

B.11 SINGULAR and MuPAD |

References (ps-file) |

Index (ps-file) |

Algorithms |

SINGULAR Examples |