From the reviews of the first edition: "It is certainly no exaggeration to say that A Singular Introduction to Commutative Algebra aims to lead a further stage in the computational revolution in commutative algebra . Among the great strengths and most distinctive features is a new, completely unified treatment of the global and local theories. making it one of the most flexible and most efficient systems of its type....another strength of Greuel and Pfister's book is its breadth of coverage of theoretical topics in the portions of commutative algebra closest to algebraic geometry, with algorithmic treatments of almost every topic....Greuel and Pfister have written a distinctive and highly useful book that should be in the library of every commutative algebraist and algebraic geometer, expert and novice alike." J.B. Little, MAA, March 2004 The second edition is substantially enlarged by a chapter on Groebner bases in non-commtative rings, a chapter on characteristic and triangular sets with applications to primary decomposition and polynomial solving and an appendix on polynomial factorization including factorization over algebraic field extensions and absolute factorization, in the uni- and multivariate case.

A Singular Introduction to Commutative Algebra

Table of basic properties Notation and basic definitions Preface 1. What is the integral closure 2. Integral closure of rings 3. Separability 4. Noetherian rings 5. Rees algebras 6. Valuations 7. Derivations 8. Reductions 9. Analytically unramified rings 10. Rees valuations 11. Multiplicity and integral closure 12. The conductor 13. The Briancon-Skoda theorem 14. Two-dimensional regular local rings 15. Computing the integral closure 16. Integral dependence of modules 17. Joint reductions 18. Adjoints of ideals 19. Normal homomorphisms Appendix A. Some background material Appendix B. Height and dimension formulas References Index.

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Integral closure of ideals, rings, and modules

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Infinite Abelian Groups

Determine the value of the variable in each equation.

Basic Algebra = 代数学入門

The complexity of the word problems for commutative semigroups and polynomial ideals

This is an introduction to, and survey of, the constructive approaches to pure mathematics The authors emphasise the viewpoint of Errett Bishop's school, but intuitionism Russian constructivism and recursive analysis are also treated, with comparisons between the various approaches included where appropriate Constructive mathematics is now enjoying a revival, with interest from not only logicans but also category theorists, recursive function theorists and theoretical computer scientists This account for non-specialists in these and other disciplines

Varieties of Constructive Mathematics

I. Sets.- 1. Constructive vs. classical mathematics.- 2. Sets, subsets and functions.- 3. Choice.- 4. Categories.- 5. Partially ordered sets and lattices.- 6. Well-founded sets and ordinals.- Notes.- II. Basic Algebra.- 1. Groups.- 2. Rings and fields.- 3. Real numbers.- 4. Modules.- 5. Polynomial rings.- 6. Matrices and vector spaces.- 7. Determinants.- 8. Symmetric polynomials.- Notes.- III. Rings And Modules.- 1. Quasi-regular ideals.- 2. Coherent and Noetherian modules.- 3. Localization.- 4. Tensor products.- 5. Flat modules.- 6. Local rings.- 7. Commutative local rings.- Notes.- IV. Divisibility in Discrete Domains.- 1. Cancellation monoids.- 2. UFD's and Bezout domains.- 3. Dedekind-Hasse rings and Euclidean domains.- 4. Polynomial rings.- Notes.- V. Principal Ideal Domains.- 1. Diagonalizing matrices.- 2. Finitely presented modules.- 3. Torsion modules, p-components, elementary divisors.- 4. Linear transformations.- Notes.- VI. Field Theory.- 1. Integral extensions and impotent rings.- 2. Algebraic independence and transcendence bases.- 3. Splitting fields and algebraic closures.- 4. Separability and diagonalizability.- 5. Primitive elements.- 6. Separability and characteristic p.- 7. Perfect fields.- 8. Galois theory.- Notes.- VII. Factoring Polynomials.- 1. Factorial and separably factorial fields.- 2. Extensions of (separably) factorial fields.- 3. Condition p.- 4. The fundamental theorem of algebra.- Notes.- VIII. Commutative Noetherian Rings.- 1. The Hilbert basis theorem.- 2. Noether normalization and the Artin-Rees lemma.- 3. The Nullstellensatz.- 4. Tennenbaum' s approach to the Hilbert basis theorem.- 5. Primary ideals.- 6. Localization.- 7. Primary decomposition.- 8. Lasker-Noether rings.- 9. Fully Lasker-Noether rings.- 10. The principal ideal theorem.- Notes.- IX. Finite Dimensional Algebras.- 1. Representations.- 2. The density theorem.- 3. The radical and summands.- 4. Wedderburn's theorem, part one.- 5. Matrix rings and division algebras.- Notes.- X. Free Groups.- 1. Existence and uniqueness.- 2. Nielsen sets.- 3. Finitely generated subgroups.- 4. Detachable subgroups of finite-rank free groups.- 5. Conjugate subgroups.- Notes.- XI. Abelian Groups.- 1. Finite-rank torsion-free groups.- 2. Divisible groups.- 3. Height functions on p-groups.- 4. Ulm's theorem.- 5. Construction of Ulm groups.- Notes.- XII. Valuation Theory.- 1. Valuations.- 2. Locally precompact valuations.- 3. Pseudofactorial fields.- 4. Normed vector spaces.- 5. Real and complex fields.- 6. Hensel's lemma.- 7. Extensions of valuations.- 8. e and f.- Notes.- XIII. Dedekind Domains.- 1. Dedekind sets of valuations.- 2. Ideal theory.- 3. Finite extensions.

A Course in Constructive Algebra

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Constructive aspects of Noetherian rings

Is it reasonable to do constructive mathematics without the axiom of countable choice? Serious schools of constructive mathematics all assume it one way or another, but the arguments for it are not compelling. The fundamental theorem of algebra will serve as an example of where countable choice comes into play and how to proceed in its absence. Along the way, a notion of a complete metric space, suitable for a choiceless environment, is developed.

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The fundamental theorem of algebra: a constructive development without choice

I was inspired, not to say provoked, to write this note by Michel J. Blais's article A pragmatic analysis of mathematical realism and intuitionism [2]. Having spent the greater part of my career doing intuitionistic mathematics, while continuing to do classical mathematics, I have come to feel that most comparisons of these two approaches to mathematics miss the essential point: intuitionism, in its simplest form, is a generalization of classical mathematics that accomodates both classical and computational models. By intuitionism I mean the approach to mathematics based on intuitionistic logic, a well-de ned body of axioms and rules of inference [6] [3]. So, for example, my idea of intuitionism does not include the notion of a choice sequence [8], or the various continuity principles associated with intuitionism [9], and it does not refer to the more bizarre consequences that have been drawn from Brouwer's idea of a creating subject [3]. This lean version of intuitionistic mathematics is usually called constructive mathematics. Blais directs his comments in [2] at constructive mathematics rather than at the more esoteric varieties of intuitionistic mathematics. Most analyses dwell on the relative merits of classical and computational models, despite the fact that constructive mathematics, viewed as a theory rather than as a description of a universe, can be interpreted as speaking about either model. Other avors of intuitionistic mathematics incorporate axioms that hold only in computational models. For example, Brouwer proved that every totally de ned function on the real line is continuous. A theory where this is provable cannot refer to the classical universe, so we have to consider whether we like its models better than the classical model. But any theorem in constructive mathematics is a theorem in classical mathematics, so in this case it is not a question of choosing between models, but of deciding whether it is worthwhile to talk about computational models in addition to the classical model. By comparing mathematical realism with intuitionism from an informal axiomatic point of view, we can steer clear of most of the metaphysical problems involved in analyzing these notions from the ground up, and concentrate on what may be termed the purely mathematical aspects. In particular we won't have to consider whether intuitionists hold that a theorem \\isn't true until it's known to be true\", as Blais

Fred Richman

Papers

Varieties of Constructive Mathematics

A Course in Constructive Algebra

Constructive aspects of Noetherian rings

The fundamental theorem of algebra: a constructive development without choice

Intuitionism As Generalization