Commutative Algebra Constructive Methods

Author : Henri Lombardi,Claude Quitté
Publisher : Springer
Page : 996 pages
File Size : 44,5 Mb
Release : 2015-07-22
Category : Mathematics
ISBN : 9789401799447

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Commutative Algebra: Constructive Methods by Henri Lombardi,Claude Quitté Pdf

Translated from the popular French edition, this book offers a detailed introduction to various basic concepts, methods, principles, and results of commutative algebra. It takes a constructive viewpoint in commutative algebra and studies algorithmic approaches alongside several abstract classical theories. Indeed, it revisits these traditional topics with a new and simplifying manner, making the subject both accessible and innovative. The algorithmic aspects of such naturally abstract topics as Galois theory, Dedekind rings, Prüfer rings, finitely generated projective modules, dimension theory of commutative rings, and others in the current treatise, are all analysed in the spirit of the great developers of constructive algebra in the nineteenth century. This updated and revised edition contains over 350 well-arranged exercises, together with their helpful hints for solution. A basic knowledge of linear algebra, group theory, elementary number theory as well as the fundamentals of ring and module theory is required. Commutative Algebra: Constructive Methods will be useful for graduate students, and also researchers, instructors and theoretical computer scientists.

Author : David Eisenbud
Publisher : Unknown
Page : 808 pages
File Size : 47,7 Mb
Release : 2014-09-01
Category : Electronic
ISBN : 1461253519

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Commutative Algebra by David Eisenbud Pdf

Author : Daniel Anderson,Ira J. Patrick
Publisher : CRC Press
Page : 378 pages
File Size : 45,8 Mb
Release : 2019-05-07
Category : Mathematics
ISBN : 9780429530449

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Ideal Theoretic Methods in Commutative Algebra by Daniel Anderson,Ira J. Patrick Pdf

Includes current work of 38 renowned contributors that details the diversity of thought in the fields of commutative algebra and multiplicative ideal theory. Summarizes recent findings on classes of going-down domains and the going-down property, emphasizing new characterizations and applications, as well as generalizations for commutative rings wi

Author : Ihsen Yengui
Publisher : Springer
Page : 277 pages
File Size : 51,6 Mb
Release : 2015-12-11
Category : Mathematics
ISBN : 9783319194943

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Constructive Commutative Algebra by Ihsen Yengui Pdf

The main goal of this book is to find the constructive content hidden in abstract proofs of concrete theorems in Commutative Algebra, especially in well-known theorems concerning projective modules over polynomial rings (mainly the Quillen-Suslin theorem) and syzygies of multivariate polynomials with coefficients in a valuation ring. Simple and constructive proofs of some results in the theory of projective modules over polynomial rings are also given, and light is cast upon recent progress on the Hermite ring and Gröbner ring conjectures. New conjectures on unimodular completion arising from our constructive approach to the unimodular completion problem are presented. Constructive algebra can be understood as a first preprocessing step for computer algebra that leads to the discovery of general algorithms, even if they are sometimes not efficient. From a logical point of view, the dynamical evaluation gives a constructive substitute for two highly nonconstructive tools of abstract algebra: the Law of Excluded Middle and Zorn's Lemma. For instance, these tools are required in order to construct the complete prime factorization of an ideal in a Dedekind ring, whereas the dynamical method reveals the computational content of this construction. These lecture notes follow this dynamical philosophy.

Author : Andrea Ferretti
Publisher : American Mathematical Society
Page : 394 pages
File Size : 46,9 Mb
Release : 2023-09-26
Category : Mathematics
ISBN : 9781470471279

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Commutative Algebra by Andrea Ferretti Pdf

This book provides an introduction to classical methods in commutative algebra and their applications to number theory, algebraic geometry, and computational algebra. The use of number theory as a motivating theme throughout the book provides a rich and interesting context for the material covered. In addition, many results are reinterpreted from a geometric perspective, providing further insight and motivation for the study of commutative algebra. The content covers the classical theory of Noetherian rings, including primary decomposition and dimension theory, topological methods such as completions, computational techniques, local methods and multiplicity theory, as well as some topics of a more arithmetic nature, including the theory of Dedekind rings, lattice embeddings, and Witt vectors. Homological methods appear in the author's sequel, Homological Methods in Commutative Algebra. Overall, this book is an excellent resource for advanced undergraduates and beginning graduate students in algebra or number theory. It is also suitable for students in neighboring fields such as algebraic geometry who wish to develop a strong foundation in commutative algebra. Some parts of the book may be useful to supplement undergraduate courses in number theory, computational algebra or algebraic geometry. The clear and detailed presentation, the inclusion of computational techniques and arithmetic topics, and the numerous exercises make it a valuable addition to any library.

Author : Wolmer Vasconcelos
Publisher : Springer Science & Business Media
Page : 432 pages
File Size : 48,6 Mb
Release : 2004-05-18
Category : Mathematics
ISBN : 3540213112

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Computational Methods in Commutative Algebra and Algebraic Geometry by Wolmer Vasconcelos Pdf

This ACM volume deals with tackling problems that can be represented by data structures which are essentially matrices with polynomial entries, mediated by the disciplines of commutative algebra and algebraic geometry. The discoveries stem from an interdisciplinary branch of research which has been growing steadily over the past decade. The author covers a wide range, from showing how to obtain deep heuristics in a computation of a ring, a module or a morphism, to developing means of solving nonlinear systems of equations - highlighting the use of advanced techniques to bring down the cost of computation. Although intended for advanced students and researchers with interests both in algebra and computation, many parts may be read by anyone with a basic abstract algebra course.

Author : Christopher Francisco,Lee C. Klingler,Sean Sather-Wagstaff,Janet C. Vassilev
Publisher : Walter de Gruyter
Page : 377 pages
File Size : 54,8 Mb
Release : 2012-04-26
Category : Mathematics
ISBN : 9783110250404

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Progress in Commutative Algebra 1 by Christopher Francisco,Lee C. Klingler,Sean Sather-Wagstaff,Janet C. Vassilev Pdf

This is the first of two volumes of a state-of-the-art survey article collection which originates from three commutative algebra sessions at the 2009 Fall Southeastern American Mathematical Society Meeting at Florida Atlantic University. The articles reach into diverse areas of commutative algebra and build a bridge between Noetherian and non-Noetherian commutative algebra. These volumes present current trends in two of the most active areas of commutative algebra: non-noetherian rings (factorization, ideal theory, integrality), and noetherian rings (the local theory, graded situation, and interactions with combinatorics and geometry). This volume contains combinatorial and homological surveys. The combinatorial papers document some of the increasing focus in commutative algebra recently on the interaction between algebra and combinatorics. Specifically, one can use combinatorial techniques to investigate resolutions and other algebraic structures as with the papers of Fløystad on Boij-Söderburg theory, of Geramita, Harbourne and Migliore, and of Cooper on Hilbert functions, of Clark on minimal poset resolutions and of Mermin on simplicial resolutions. One can also utilize algebraic invariants to understand combinatorial structures like graphs, hypergraphs, and simplicial complexes such as in the paper of Morey and Villarreal on edge ideals. Homological techniques have become indispensable tools for the study of noetherian rings. These ideas have yielded amazing levels of interaction with other fields like algebraic topology (via differential graded techniques as well as the foundations of homological algebra), analysis (via the study of D-modules), and combinatorics (as described in the previous paragraph). The homological articles the editors have included in this volume relate mostly to how homological techniques help us better understand rings and singularities both noetherian and non-noetherian such as in the papers by Roberts, Yao, Hummel and Leuschke.

Author : David Eisenbud
Publisher : Springer Science & Business Media
Page : 784 pages
File Size : 52,9 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9781461253501

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Commutative Algebra by David Eisenbud Pdf

This is a comprehensive review of commutative algebra, from localization and primary decomposition through dimension theory, homological methods, free resolutions and duality, emphasizing the origins of the ideas and their connections with other parts of mathematics. The book gives a concise treatment of Grobner basis theory and the constructive methods in commutative algebra and algebraic geometry that flow from it. Many exercises included.

Author : Andrea Ferretti
Publisher : American Mathematical Society
Page : 432 pages
File Size : 42,7 Mb
Release : 2023-12-05
Category : Mathematics
ISBN : 9781470474362

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Homological Methods in Commutative Algebra by Andrea Ferretti Pdf

This book develops the machinery of homological algebra and its applications to commutative rings and modules. It assumes familiarity with basic commutative algebra, for example, as covered in the author's book, Commutative Algebra. The first part of the book is an elementary but thorough exposition of the concepts of homological algebra, starting from categorical language up to the construction of derived functors and spectral sequences. A full proof of the celebrated Freyd-Mitchell theorem on the embeddings of small Abelian categories is included. The second part of the book is devoted to the application of these techniques in commutative algebra through the study of projective, injective, and flat modules, the construction of explicit resolutions via the Koszul complex, and the properties of regular sequences. The theory is then used to understand the properties of regular rings, Cohen-Macaulay rings and modules, Gorenstein rings and complete intersections. Overall, this book is a valuable resource for anyone interested in learning about homological algebra and its applications in commutative algebra. The clear and thorough presentation of the material, along with the many examples and exercises of varying difficulty, make it an excellent choice for self-study or as a reference for researchers.

Author : Thomas Becker,Volker Weispfenning
Publisher : Springer Science & Business Media
Page : 587 pages
File Size : 44,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9781461209133

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Gröbner Bases by Thomas Becker,Volker Weispfenning Pdf

The origins of the mathematics in this book date back more than two thou sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbek istan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al gorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations. During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method.

Author : J. Alajbegovic,J. Mockor
Publisher : Springer Science & Business Media
Page : 339 pages
File Size : 53,6 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401127165

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Approximation Theorems in Commutative Algebra by J. Alajbegovic,J. Mockor Pdf

Various types of approximation theorems are frequently used in general commutative algebra, and they have been found to be useful tools in valuation theory, the theory of Abelian lattice ordered groups, multiplicative ideal theory, etc. Part 1 of this volume is devoted to the investigation of approximation theorems from a classical point of view. The chapters of this part deal with fields and rings, partly ordered groups, and with multirings and d-groups. Part II investigates approximation theorems from a general, categorical point of view. This part is essentially self-contained and requires only a basic knowledge of category theory and first-order logic. For researchers and graduate students of commutative algebra, category theory, as well as applications of logic.

Author : J.L. Bueso,José Gómez-Torrecillas,A. Verschoren
Publisher : Springer Science & Business Media
Page : 307 pages
File Size : 44,9 Mb
Release : 2013-03-09
Category : Computers
ISBN : 9789401702850

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Algorithmic Methods in Non-Commutative Algebra by J.L. Bueso,José Gómez-Torrecillas,A. Verschoren Pdf

The already broad range of applications of ring theory has been enhanced in the eighties by the increasing interest in algebraic structures of considerable complexity, the so-called class of quantum groups. One of the fundamental properties of quantum groups is that they are modelled by associative coordinate rings possessing a canonical basis, which allows for the use of algorithmic structures based on Groebner bases to study them. This book develops these methods in a self-contained way, concentrating on an in-depth study of the notion of a vast class of non-commutative rings (encompassing most quantum groups), the so-called Poincaré-Birkhoff-Witt rings. We include algorithms which treat essential aspects like ideals and (bi)modules, the calculation of homological dimension and of the Gelfand-Kirillov dimension, the Hilbert-Samuel polynomial, primality tests for prime ideals, etc.

Author : Thomas Becker,Volker Weispfenning
Publisher : Springer
Page : 576 pages
File Size : 46,9 Mb
Release : 1998-03-23
Category : Mathematics
ISBN : 0387979719

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Gröbner Bases by Thomas Becker,Volker Weispfenning Pdf

The origins of the mathematics in this book date back more than two thou sand years, as can be seen from the fact that one of the most important algorithms presented here bears the name of the Greek mathematician Eu clid. The word "algorithm" as well as the key word "algebra" in the title of this book come from the name and the work of the ninth-century scientist Mohammed ibn Musa al-Khowarizmi, who was born in what is now Uzbek istan and worked in Baghdad at the court of Harun al-Rashid's son. The word "algorithm" is actually a westernization of al-Khowarizmi's name, while "algebra" derives from "al-jabr," a term that appears in the title of his book Kitab al-jabr wa'l muqabala, where he discusses symbolic methods for the solution of equations. This close connection between algebra and al gorithms lasted roughly up to the beginning of this century; until then, the primary goal of algebra was the design of constructive methods for solving equations by means of symbolic transformations. During the second half of the nineteenth century, a new line of thought began to enter algebra from the realm of geometry, where it had been successful since Euclid's time, namely, the axiomatic method.

Author : Alexander Martsinkovsky
Publisher : Springer Nature
Page : 256 pages
File Size : 55,8 Mb
Release : 2024-07-01
Category : Electronic
ISBN : 9783031530630

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Functor Categories, Model Theory, Algebraic Analysis and Constructive Methods by Alexander Martsinkovsky Pdf

Author : Ray Mines,Fred Richman,Wim Ruitenburg
Publisher : Springer Science & Business Media
Page : 355 pages
File Size : 46,6 Mb
Release : 2012-09-10
Category : Mathematics
ISBN : 9781441986405

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A Course in Constructive Algebra by Ray Mines,Fred Richman,Wim Ruitenburg Pdf

The constructive approach to mathematics has enjoyed a renaissance, caused in large part by the appearance of Errett Bishop's book Foundations of constr"uctiue analysis in 1967, and by the subtle influences of the proliferation of powerful computers. Bishop demonstrated that pure mathematics can be developed from a constructive point of view while maintaining a continuity with classical terminology and spirit; much more of classical mathematics was preserved than had been thought possible, and no classically false theorems resulted, as had been the case in other constructive schools such as intuitionism and Russian constructivism. The computers created a widespread awareness of the intuitive notion of an effecti ve procedure, and of computation in principle, in addi tion to stimulating the study of constructive algebra for actual implementation, and from the point of view of recursive function theory. In analysis, constructive problems arise instantly because we must start with the real numbers, and there is no finite procedure for deciding whether two given real numbers are equal or not (the real numbers are not discrete) . The main thrust of constructive mathematics was in the direction of analysis, although several mathematicians, including Kronecker and van der waerden, made important contributions to construc tive algebra. Heyting, working in intuitionistic algebra, concentrated on issues raised by considering algebraic structures over the real numbers, and so developed a handmaiden'of analysis rather than a theory of discrete algebraic structures.