The Algebraic Theory of Modular Systems (Classic Reprint) - Couverture souple

Francis Sowerby Macaulay

Couverture souple

ISBN 10 : 1330386671 ISBN 13 : 9781330386675

Editeur : Forgotten Books, 2018

Synopsis

Explore the algebraic foundations behind modular systems and their applications. This concise, theory‑driven work guides you through ideals, modules, and the relationships that bind them, from basic definitions to the structure of complex systems.

This edition presents a clear account of how rank, unmixed versus mixed modules, and prime factors interact within modular systems. It also contrasts Kronecker’s and Dedekind’s approaches, and shows how results can be used to understand linear equations and their solutions in an algebraic setting.

• Understand what it means for a module or ideal to have a certain rank and when it is unmixed.


• Learn how prime factors and divisibility operate in the context of modular systems and algebraic forms.


• See how the theory connects to solving homogeneous linear equations and to the concept of resultants.


• Compare classical viewpoints from Kronecker and Dedekind to modern interpretations.

Ideal for readers who want a rigorous, historically grounded look at the algebra of modular systems and the tools used to factor and solve within that framework.

A propos du livre

Présentation de l'éditeur

From the Preface.
THE present state of our knowledge of the properties of Modular Systems is chiefly due to the fundamental theorems and processes of L. Kronecker, M. Noether, D. Hilbert, and E. Lasker, and above all to J. Konig's profound exposition and numerous extensions of Kronecker's theory (p. xiii). Konig's treatise might be regarded as in some measure complete if it were admitted that a problem is finished with when its solution has been reduced to a finite number of feasible operations. If however the operations are too numerous or too involved to be carried out in practice the solution is only a theoretical one; and its importance then lies not in itself, but in the theorems with which it is associated and to which it leads. Such a theoretical solution must be regarded as a preliminary and not the final stage in the consideration of the problem.
In the following presentment of the subject Section I is devoted to the Resultant, the case of equations being treated in a parallel manner to that of two equations; Section II contains an account of Kronecker's theory of the Resolvent, following mainly the lines of Konig's exposition ; Section III, on general properties, is closely allied to Lasker's memoir and Dedekind's theory of Ideals; and Section IV is an extension of Lasker's results founded on the methods originated by Noether. The additions to the theory consist of one or two isolated theorems (especially §§ 50 — 53 and § 79 and its consequences) and the introduction of the Inverse System in Section IV.
The subject is full of pitfalls. I have pointed out some mistakes made by others, but have no doubt that I have made new ones. It may be expected that any errors will be discovered and eliminated in due course, since proofs or references are given for all major and most minor statements. I take this opportunity of thanking the Editors for their acceptance of this tract and the Syndics of the University Press for publishing it.

Présentation de l'éditeur

Many of the ideas introduced by Macaulay in this book have developed into central concepts in what has become the branch of mathematics known as Commutative Algebra. Today his name is remembered through the term 'Cohen–Macaulay ring', however, it is less well known that he pioneered several other fundamental ideas, including the concept of the Gorenstein ring and the use of injective modules, ideas which were not systematically developed until considerably later in this century. An introduction by Professor Paul Roberts links past with present. The background to Macaulay's thinking is discussed, and the development of modern theory is outlined. The wealth of ideas expounded here by Macaulay over 75 years ago, will still be a source of inspiration to all workers in commutative algebra.

Détails de l'édition

• Éditeur: Forgotten Books
• Date d'édition: 2018
• Langue: anglais
• ISBN 10: 1330386671
• ISBN 13: 9781330386675
• Reliure: Broché
• Nombre de pages: 133
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