Mathematics

The primary audience for this book is students and the young researchers interested in the core of the discipline. Commutative algebra is by and large a self-contained discipline, which makes it quite dry for the beginner with a basic training in elementary algebra and calculus.

A stable mathematical discipline such as this enshrines a vital number of topics to be learned at an early stage, more or less universally accepted and practiced. Naturally, authors tend to turn these topics into an increasingly short and elegant list of basic facts of the theory. So, the shorter the better. However, there is a subtle watershed between elegance and usefulness, especially if the target is the beginner. From my experience throughout years of teaching, elegance and terseness do not do it, except much later in the carrier. To become useful, the material ought to carry quite a bit of motivation through justification and usefulness pointers.

On the other hand, it is difficult to contemplate these teaching devices in the writing of a short book. I have divided the material in three parts. starting with more elementary sections, then carrying an intermezzo on more difficult themes to make up for a smooth crescendo with additional tools and, finally, the more advanced part, versing on a reasonable chunk of present-day steering of commutative algebra.

Historic notes at the end of each chapter provide insight into the original sources and background information on a particular subject or theorem.

Exercises are provided and propose problems that apply the theory to solve concrete questions (yes, with concrete polynomials, and so forth).

This monograph is devoted to various types of algebras of functions with n variables. It is the first complete monograph (in English) on this topic, covering mainly the Russian literature. It is addressed to all algebraists working in the area of universal algebras, semigroup theory, etc. It is also a useful source of information for graduate and PhD students who are starting their research in this area.

The book is the first monograph in the English mathematical literature which provides readers with a very systematical study of the notion of Menger algebras, and its generalizations and applications. The results presented here were originally published mostly in the Russian literature: In 2006, the first version of this book was edited in Russian and it is now presented in an extended version, where two new and very important chapters are added. The monograph is a broad survey of unknown or little-known Russian literature on algebras of multiplace functions and presents to the mathematical community a beautiful and strongly developing theory.

Dieses Lehrbuch bietet eine Einführung in die grundlegenden Methoden der Galoistheorie. Am Beispiel der Auflösbarkeit von Polynomgleichungen durch Radikale wird das Zusammenwirken dreier Theorien – Gruppentheorie, Körpertheorie und Ringtheorie – zur Lösung dieses Problems demonstriert. Behandelt werden neben den üblichen Grundbegriffen wie Gruppen, Körper und Ringe sowie den Resultaten der Galoistheorie auch Anwendungen auf Konstruktionen mit Zirkel und Lineal, endliche Körper und Kreisteilungskörper sowie Auflösungsformeln der Gleichungen vom Grad höchstens 4. Darüber hinaus wird der konkreten Berechenbarkeit und den Algorithmen zur Bestimmung irreduzibler Teiler von Polynomen bzw. der Galoisgruppe eines moderaten Polynoms ein breiter Raum gewidmet.

Die vorliegende zweite Auflage enthält Erweiterungen zu den Themen rein inseparable Körpererweiterungen, p-adische Zahlen und Bewertungstheorie, angeordnete Körper und Satz von Sturm.

The category of all modules over a general associative ring is too complex to admit any reasonable classification. Thus, unless the ring is of finite representation type, one must limit attempts at classification to some restricted subcategories of modules.

The wild character of the category of all modules, or of one of its subcategories C is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions and these are generally viewed as obstacles to the classification. Realization theorems have thus become important indicators of the non-classification theory of modules.

In order to overcome this problem, approximation theory of modules has been developed over the past few decades. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by ones from C. These approximations are neither unique nor functorial in general, but there is always a rich supply available appropriate to the requirements of various particular applications. Thus approximation theory has developed into an important part of the classification theory of modules.

In this monograph the two methods are brought together. First the approximation theory of modules is developed and some of its recent applications, notably to infinite dimensional tilting theory, are presented. Then some prediction principles from set theory are introduced and these become the principal tools in the establishment of appropriate realization theorems.

The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

This proceedings volume documents the contributions presented at the CONIAPS XXVII International Conference on Recent Advances in Pure and Applied Algebra. The entries focus on modern trends and techniques in various branches of pure and applied Algebra and highlight their applications in coding, cryptography, graph, and fuzzy theory. The book comprised a total of eighteen chapters, among which the first fourteen chapters are devoted to Algebra and related topics, and the last four chapters are included applied mathematics parts. The chapters present the latest research work being done on the frontiers of the various branches of algebra as well as showcase the cross-fertilization of the ideas and connection among these branches.Covering a broad range of topics in pure and applied Algebra, this volume would appeal to a wide spectrum of the researcher in Mathematics. The main aim of this monograph is to contribute to the development of pure and applied Algebra and hence we purposely sought a cross-section of topics in Algebra and encouraged expository presentations and research papers that provide an innovative link between research areas of Algebra and the field of their applications. This volume will be useful not only to experts but also to beginners of research in algebras and related topics.

This title from the De Gruyter Book Archive has been digitized in order to make it available for academic research. It was originally published under National Socialism and has to be viewed in this historical context. Learn more here.

In a self contained and exhaustive work the author covers Group Theory in its multifaceted aspects, treating its conceptual foundations in a proper logical order. First discrete and finite group theory, that includes the entire chemical-physical field of crystallography is developed self consistently, followed by the structural theory of Lie Algebras with a complete exposition of the roots and Dynkin diagrams lore. A primary on Fibre-Bundles, Connections and Gauge fields, Riemannian Geometry and the theory of Homogeneous Spaces G/H is also included and systematically developed.

https://petrusfremathandlit.net

This second, revised and substantially extended edition of Approximations and Endomorphism Algebras of Modules reflects both the depth and the width of recent developments in the area since the first edition appeared in 2006. The new division of the monograph into two volumes roughly corresponds to its two central topics, approximation theory (Volume 1) and realization theorems for modules (Volume 2).

It is a widely accepted fact that the category of all modules over a general associative ring is too complex to admit classification. Unless the ring is of finite representation type we must limit attempts at classification to some restricted subcategories of modules. The wild character of the category of all modules, or of one of its subcategories C, is often indicated by the presence of a realization theorem, that is, by the fact that any reasonable algebra is isomorphic to the endomorphism algebra of a module from C. This results in the existence of pathological direct sum decompositions, and these are generally viewed as obstacles to classification. In order to overcome this problem, the approximation theory of modules has been developed. The idea here is to select suitable subcategories C whose modules can be classified, and then to approximate arbitrary modules by those from C. These approximations are neither unique nor functorial in general, but there is a rich supply available appropriate to the requirements of various particular applications.

The authors bring the two theories together. The first volume, Approximations, sets the scene in Part I by introducing the main classes of modules relevant here: the S-complete, pure-injective, Mittag-Leffler, and slender modules. Parts II and III of the first volume develop the key methods of approximation theory. Some of the recent applications to the structure of modules are also presented here, notably for tilting, cotilting, Baer, and Mittag-Leffler modules. In the second volume, Predictions, further basic instruments are introduced: the prediction principles, and their applications to proving realization theorems. Moreover, tools are developed there for answering problems motivated in algebraic topology. The authors concentrate on the impossibility of classification for modules over general rings. The wild character of many categories C of modules is documented here by the realization theorems that represent critical R-algebras over commutative rings R as endomorphism algebras of modules from C.

The monograph starts from basic facts and gradually develops the theory towards its present frontiers. It is suitable both for graduate students interested in algebra and for experts in module and representation theory.

This volume is the proceedings of the conference on Automorphic Representations, L-functions and Applications: Progress and Prospects, held at the Department of Mathematics of The Ohio State University, March 27–30, 2003, in honor of the 60th birthday of Steve Rallis.

The theory of automorphic representations, automorphic L-functions and their applications to arithmetic continues to be an area of vigorous and fruitful research. The contributed papers in this volume represent many of the most recent developments and directions, including

• Rankin–Selberg L-functions (Bump, Ginzburg–Jiang–Rallis, Lapid–Rallis)
• the relative trace formula (Jacquet, Mao–Rallis)
• automorphic representations (Gan–Gurevich, Ginzburg–Rallis–Soudry)
• representation theory of p-adic groups (Baruch, Kudla–Rallis, Mœglin, Cogdell–Piatetski-Shapiro–Shahidi)
• p-adic methods (Harris–Li–Skinner, Vigneras), and
• arithmetic applications (Chinta–Friedberg–Hoffstein).

The survey articles by Bump, on the Rankin–Selberg method, and by Jacquet, on the relative trace formula, should be particularly useful as an introduction to the key ideas about these important topics.

This volume should be of interest both to researchers and students in the area of automorphic representations, as well as to mathematicians in other areas interested in having an overview of current developments in this important field.

Jordanus de Nemore's De Numeris Datis stands as a groundbreaking text, marking the advent of advanced algebra in Western Europe. Written in the 13th century, the work assumes mastery of basic algebra and progresses to develop a systematic approach to quadratic, simultaneous, and proportional equations, many of which had not been previously articulated in European mathematics. By providing sophisticated techniques for numerical analysis, Jordanus not only addressed the needs of his contemporary mathematical community but also laid a foundational stone for the algebraic advancements that followed. The text bridges a gap between the classical mathematical traditions of the Greeks and the revolutionary algebraic methods emerging in the medieval Islamic world, showcasing Jordanus’s role as an intellectual conduit in a period of vibrant cross-cultural exchange.

Despite scant biographical information about Jordanus himself, his contributions resonate through the depth and structure of his work. De Numeris Datis combines theoretical rigor with practical application, offering propositions and methods that are both abstract and directly applicable to real-world problems. The text's influence is underscored by its enduring presence in medieval and Renaissance scholarship, with numerous surviving manuscripts and commentaries attesting to its relevance. By formalizing algebraic analysis in a way that prefigures later works like Viète's Introduction to the Analytical Art, Jordanus elevated algebra from a computational tool to a discipline of logical inquiry, cementing his legacy as one of the seminal figures in the history of mathematics.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1981.

Jordanus de Nemore's De Numeris Datis stands as a groundbreaking text, marking the advent of advanced algebra in Western Europe. Written in the 13th century, the work assumes mastery of basic algebra and progresses to develop a systematic approach to quadratic, simultaneous, and proportional equations, many of which had not been previously articulated in European mathematics. By providing sophisticated techniques for numerical analysis, Jordanus not only addressed the needs of his contemporary mathematical community but also laid a foundational stone for the algebraic advancements that followed. The text bridges a gap between the classical mathematical traditions of the Greeks and the revolutionary algebraic methods emerging in the medieval Islamic world, showcasing Jordanus’s role as an intellectual conduit in a period of vibrant cross-cultural exchange.

Despite scant biographical information about Jordanus himself, his contributions resonate through the depth and structure of his work. De Numeris Datis combines theoretical rigor with practical application, offering propositions and methods that are both abstract and directly applicable to real-world problems. The text's influence is underscored by its enduring presence in medieval and Renaissance scholarship, with numerous surviving manuscripts and commentaries attesting to its relevance. By formalizing algebraic analysis in a way that prefigures later works like Viète's Introduction to the Analytical Art, Jordanus elevated algebra from a computational tool to a discipline of logical inquiry, cementing his legacy as one of the seminal figures in the history of mathematics.

This title is part of UC Press's Voices Revived program, which commemorates University of California Press’s mission to seek out and cultivate the brightest minds and give them voice, reach, and impact. Drawing on a backlist dating to 1893, Voices Revived makes high-quality, peer-reviewed scholarship accessible once again using print-on-demand technology. This title was originally published in 1981.