Preface (psfile) 

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 Zerodimensional 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 HilbertPoincare Series 
5.3 Properties of the Hilbert Polynomial 
5.4 Filtrations and the Lemma of ArtinRees 
5.5 The HilbertSamuel 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 CohenMacaulay Rings 
7.8 Further Characterization of CohenMacaulayness 
A. Geometric Background 
A.1 Introduction by Pictures (psfile) 
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 (psfile) 
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 (psfile) 
Index (psfile) 
Algorithms 
SINGULAR Examples
SINGULAR Examples
algebraic dependence, 87 
annihilator, 186 

Betti numbers, 135 
 graded, 137 

classification of singularities, 493 
computation 
 in fields, 5 
 in polynomial rings, 7 
 in quotient rings, 25 
 of d(I,K[x]), 222 
 of Hom, 106 
 of the dimension, 211 
 of Tor, 340 
computing with radicals, 27 
counting nodes, 489 
creating ring maps, 8 
cyclic decomposition, 159 

deformation of singularities, 495 
degree, 289 
 of projection, 469 
 of projective variety, 478 
diagonal form, 154 
dimension, 289 
 embedding, 304 
 of a module, 302 

elimination 
 and resultant, 431 
 of module components, 180 
 of variables, 71 
 projective, 466 
equidimensional 
 decomposition, 263 
 part, 261 
estimating the determinacy, 491 

finite maps, 196 
finiteness test, 324 
Fitting ideal, 186, 345 
flat locus, 356 
flatness test, 369 
flattening stratification, 352 

global versus local rings, 35 
graded 
 Betti numbers, 137 
 rings and modules, 116 

highest corner, 60 
Hilbert 
 function, 289 
 polynomial, 299 
HilbertPoincare series, 282 
homogeneous resolution, 137 

ideal membership, 68 
image of module homomorphism, 99 
independent set, 220 
initial ideal, 299 
injective, 420 
integral 
 closure of an ideal, 202 
 elements, 195 
intersection 
 of ideals, 79 
 of submodules, 102, 181 
inverse of a power series, 316 

Jacobian criterion, 304 
Jordan normal form, 163 

kernel 
 of a ring map, 85 
 of module homomorphism, 99, 187 
Koszul complex, 378 

leading data, 11 
linear combination of ideal members, 68 
local and global dimension, 472 
lying over theorem, 225 

maps induced by Hom, 96 
matrix operations, 94 
Milnor and Tjurina number, 488 
minimal 
 associated primes, 209 
 presentations, 109 
module 
 annihilator, 186 
 membership, 178 
 presentation of, 104 
 quotient, 102 
 radical and zerodivisors, 184 
monomial orderings, 16 
morphisms of projective varieties, 455 
multiplicity, 480 

Noether normalization, 216 
nonnormal locus, 232 
normal form, 51, 123 
normalization, 230 

Poincare series, 299 
presentation of a module, 104 
primary 
 decomposition, 258 
 test, 252 
projective 
 closure, 443 
 elimination, 466 
 Nullstellensatz, 437 
 subschemes, 446 
properties of ring maps, 20 

quotient 
 of ideals, 81 
 of submodules, 102, 183 

radical, 184, 265 
 membership, 78 
realization of rings, 42 
reduction to zerodimensional case, 257 
regular 
 sequences, 372 
 system of parameters, 304 
regularity test, 400 
resolution, 135 
 homogeneous, 137 

saturation, 83, 446 
Schreyer resolution, 150 
singular locus, 309 
solving equations, 76 
 linear, 189 
standard bases, 59, 124 
subalgebra membership, 87 
submodules, 104 
 intersection of, 102, 181 
 of A^{n}, 98 
sum of submodules, 102 
surface plot, 407, 413 
surjective, 420 
syzygies, 141 

tangent cone, 480 
tensor product 
 of maps, 170 
 of modules, 172 
 of rings, 175 
test 
 for CohenMacaulayness, 386, 392, 394 
 for flatness, 354 
 for local freeness, 347 

Weierstrass polynomial, 321 

Zariski closure of the image, 74 
zerodimensional primary decomposition, 253 
zerodivisors, 184 
zgeneral power series, 320 
