Lattice Ordered Rings And Modules

Author : Stuart A. Steinberg
Publisher : Springer Science & Business Media
Page : 639 pages
File Size : 48,5 Mb
Release : 2009-11-19
Category : Mathematics
ISBN : 9781441917218

Get Book

Lattice-ordered Rings and Modules by Stuart A. Steinberg Pdf

This book provides an exposition of the algebraic aspects of the theory of lattice-ordered rings and lattice-ordered modules. All of the background material on rings, modules, and lattice-ordered groups necessary to make the work self-contained and accessible to a variety of readers is included. Filling a gap in the literature, Lattice-Ordered Rings and Modules may be used as a textbook or for self-study by graduate students and researchers studying lattice-ordered rings and lattice-ordered modules. Steinberg presents the material through 800+ extensive examples of varying levels of difficulty along with numerous exercises at the end of each section. Key topics include: lattice-ordered groups, rings, and fields; archimedean $l$-groups; f-rings and larger varieties of $l$-rings; the category of f-modules; various commutativity results.

Author : Jingjing Ma
Publisher : World Scientific
Page : 260 pages
File Size : 42,6 Mb
Release : 2014-03-14
Category : Mathematics
ISBN : 9789814571449

Get Book

Lecture Notes On Algebraic Structure Of Lattice-ordered Rings by Jingjing Ma Pdf

Algebraic Structure of Lattice-Ordered Rings presents an introduction to the theory of lattice-ordered rings and some new developments in this area in the last 10-15 years. It aims to provide the reader with a good foundation in the subject, as well as some new research ideas and topic in the field.This book may be used as a textbook for graduate and advanced undergraduate students who have completed an abstract algebra course including general topics on group, ring, module, and field. It is also suitable for readers with some background in abstract algebra and are interested in lattice-ordered rings to use as a self-study book.The book is largely self-contained, except in a few places, and contains about 200 exercises to assist the reader to better understand the text and practice some ideas.

Author : K. Glazek
Publisher : Springer Science & Business Media
Page : 394 pages
File Size : 51,8 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9789401599641

Get Book

A Guide to the Literature on Semirings and their Applications in Mathematics and Information Sciences by K. Glazek Pdf

This volume presents a short guide to the extensive literature concerning semir ings along with a complete bibliography. The literature has been created over many years, in variety of languages, by authors representing different schools of mathematics and working in various related fields. In many instances the terminology used is not universal, which further compounds the difficulty of locating pertinent sources even in this age of the Internet and electronic dis semination of research results. So far there has been no single reference that could guide the interested scholar or student to the relevant publications. This book is an attempt to fill this gap. My interest in the theory of semirings began in the early sixties, when to gether with Bogdan W ~glorz I tried to investigate some algebraic aspects of compactifications of topological spaces, semirings of semicontinuous functions, and the general ideal theory for special semirings. (Unfortunately, local alge braists in Poland told me at that time that there was nothing interesting in investigating semiring theory because ring theory was still being developed). However, some time later we became aware of some similar investigations hav ing already been done. The theory of semirings has remained "my first love" ever since, and I have been interested in the results in this field that have been appearing in literature (even though I have not been active in this area myself).

Author : Carl Faith
Publisher : Springer Science & Business Media
Page : 589 pages
File Size : 48,9 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642806346

Get Book

Algebra by Carl Faith Pdf

VI of Oregon lectures in 1962, Bass gave simplified proofs of a number of "Morita Theorems", incorporating ideas of Chase and Schanuel. One of the Morita theorems characterizes when there is an equivalence of categories mod-A R::! mod-B for two rings A and B. Morita's solution organizes ideas so efficiently that the classical Wedderburn-Artin theorem is a simple consequence, and moreover, a similarity class [AJ in the Brauer group Br(k) of Azumaya algebras over a commutative ring k consists of all algebras B such that the corresponding categories mod-A and mod-B consisting of k-linear morphisms are equivalent by a k-linear functor. (For fields, Br(k) consists of similarity classes of simple central algebras, and for arbitrary commutative k, this is subsumed under the Azumaya [51]1 and Auslander-Goldman [60J Brauer group. ) Numerous other instances of a wedding of ring theory and category (albeit a shot gun wedding!) are contained in the text. Furthermore, in. my attempt to further simplify proofs, notably to eliminate the need for tensor products in Bass's exposition, I uncovered a vein of ideas and new theorems lying wholely within ring theory. This constitutes much of Chapter 4 -the Morita theorem is Theorem 4. 29-and the basis for it is a corre spondence theorem for projective modules (Theorem 4. 7) suggested by the Morita context. As a by-product, this provides foundation for a rather complete theory of simple Noetherian rings-but more about this in the introduction.

Author : V.M. Kopytov,N.Ya. Medvedev
Publisher : Springer Science & Business Media
Page : 408 pages
File Size : 41,5 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9789401583046

Get Book

The Theory of Lattice-Ordered Groups by V.M. Kopytov,N.Ya. Medvedev Pdf

A partially ordered group is an algebraic object having the structure of a group and the structure of a partially ordered set which are connected in some natural way. These connections were established in the period between the end of 19th and beginning of 20th century. It was realized that ordered algebraic systems occur in various branches of mathemat ics bound up with its fundamentals. For example, the classification of infinitesimals resulted in discovery of non-archimedean ordered al gebraic systems, the formalization of the notion of real number led to the definition of ordered groups and ordered fields, the construc tion of non-archimedean geometries brought about the investigation of non-archimedean ordered groups and fields. The theory of partially ordered groups was developed by: R. Dedekind, a. Holder, D. Gilbert, B. Neumann, A. I. Mal'cev, P. Hall, G. Birkhoff. These connections between partial order and group operations allow us to investigate the properties of partially ordered groups. For exam ple, partially ordered groups with interpolation property were intro duced in F. Riesz's fundamental paper [1] as a key to his investigations of partially ordered real vector spaces, and the study of ordered vector spaces with interpolation properties were continued by many functional analysts since. The deepest and most developed part of the theory of partially ordered groups is the theory of lattice-ordered groups. In the 40s, following the publications of the works by G. Birkhoff, H. Nakano and P.

Author : M.E Anderson,T.H. Feil
Publisher : Springer Science & Business Media
Page : 197 pages
File Size : 42,8 Mb
Release : 2012-12-06
Category : Computers
ISBN : 9789400928718

Get Book

Lattice-Ordered Groups by M.E Anderson,T.H. Feil Pdf

The study of groups equipped with a compatible lattice order ("lattice-ordered groups" or "I!-groups") has arisen in a number of different contexts. Examples of this include the study of ideals and divisibility, dating back to the work of Dedekind and continued by Krull; the pioneering work of Hahn on totally ordered abelian groups; and the work of Kantorovich and other analysts on partially ordered function spaces. After the Second World War, the theory of lattice-ordered groups became a subject of study in its own right, following the publication of fundamental papers by Birkhoff, Nakano and Lorenzen. The theory blossomed under the leadership of Paul Conrad, whose important papers in the 1960s provided the tools for describing the structure for many classes of I!-groups in terms of their convex I!-subgroups. A particularly significant success of this approach was the generalization of Hahn's embedding theorem to the case of abelian lattice-ordered groups, work done with his students John Harvey and Charles Holland. The results of this period are summarized in Conrad's "blue notes" [C].

Author : Grigore Calugareanu
Publisher : Springer Science & Business Media
Page : 233 pages
File Size : 41,8 Mb
Release : 2013-04-17
Category : Mathematics
ISBN : 9789401595889

Get Book

Lattice Concepts of Module Theory by Grigore Calugareanu Pdf

It became more and more usual, from, say, the 1970s, for each book on Module Theory, to point out and prove some (but in no more than 15 to 20 pages) generalizations to (mostly modular) lattices. This was justified by the nowadays widely accepted perception that the structure of a module over a ring is best understood in terms of the lattice struc ture of its submodule lattice. Citing Louis H. Rowen "this important example (the lattice of all the submodules of a module) is the raison d'etre for the study of lattice theory by ring theorists". Indeed, many module-theoretic results can be proved by using lattice theory alone. The purpose of this book is to collect and present all and only the results of this kind, although for this purpose one must develop some significant lattice theory. The results in this book are of the following categories: the folklore of Lattice Theory (to be found in each Lattice Theory book), module theoretic results generalized in (modular, and possibly compactly gen erated) lattices (to be found in some 6 to 7 books published in the last 20 years), very special module-theoretic results generalized in lattices (e. g. , purity in Chapter 9 and several dimensions in Chapter 13, to be found mostly in [27], respectively, [34] and [18]) and some new con cepts (e. g.

Author : Joachim Lambek
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 42,8 Mb
Release : 2009
Category : Associative rings
ISBN : 9780821849002

Get Book

Lectures on Rings and Modules by Joachim Lambek Pdf

This book is an introduction to the theory of associative rings and their modules, designed primarily for graduate students. The standard topics on the structure of rings are covered, with a particular emphasis on the concept of the complete ring of quotients. A survey of the fundamental concepts of algebras in the first chapter helps to make the treatment self-contained. The topics covered include selected results on Boolean and other commutative rings, the classical structure theory of associative rings, injective modules, and rings of quotients. The final chapter provides an introduction to homological algebra. Besides three appendices on further results, there is a six-page section of historical comments. Table of Contents: Fundamental Concepts of Algebra: 1.1 Rings and related algebraic systems; 1.2 Subrings, homomorphisms, ideals; 1.3 Modules, direct products, and direct sums; 1.4 Classical isomorphism theorems. Selected Topics on Commutative Rings: 2.1 Prime ideals in commutative rings; 2.2 Prime ideals in special commutative rings; 2.3 The complete ring of quotients of a commutative ring; 2.4 Rings of quotients of commutative semiprime rings; 2.5 Prime ideal spaces.Classical Theory of Associative Rings: 3.1 Primitive rings; 3.2 Radicals; 3.3 Completely reducible modules; 3.4 Completely reducible rings; 3.5 Artinian and Noetherian rings; 3.6 On lifting idempotents; 3.7 Local and semiperfect rings. Injectivity and Related Concepts: 4.1 Projective modules; 4.2 Injective modules; 4.3 The complete ring of quotients; 4.4 Rings of endomorphisms of injective modules; 4.5 Regular rings of quotients; 4.6 Classical rings of quotients; 4.7 The Faith-Utumi theorem. Introduction to Homological Algebra: 5.1 Tensor products of modules; 5.2 Hom and $\otimes$ as functors; 5.3 Exact sequences; 5.4 Flat modules; 5.5 Torsion and extension products. Appendixes; Comments; Bibliography; Index. Review from Zentralblatt Math: Due to their clarity and intelligible presentation, these lectures on rings and modules are a particularly successful introduction to the surrounding circle of ideas. Review from American Mathematical Monthly: An introduction to associative rings and modules which requires of the reader only the mathematical maturity which one would attain in a first-year graduate algebra [course]...in order to make the contents of the book as accessible as possible, the author develops all the fundamentals he will need.In addition to covering the basic topics...the author covers some topics not so readily available to the nonspecialist...the chapters are written to be as independent as possible...[which will be appreciated by] students making their first acquaintance with the subject...one of the most successful features of the book is that it can be read by graduate students with little or no help from a specialist. (CHEL/283.H)

Author : Jorge Martínez
Publisher : Springer Science & Business Media
Page : 288 pages
File Size : 55,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789400924727

Get Book

Ordered Algebraic Structures by Jorge Martínez Pdf

Proceedings of the Caribbean Mathematics Foundation Conference, held in Curaçao, August 1988

Author : M. Dickmann,F. Miraglia
Publisher : American Mathematical Soc.
Page : 129 pages
File Size : 55,6 Mb
Release : 2015-10-27
Category : Commutative rings
ISBN : 9781470414689

Get Book

Faithfully Quadratic Rings by M. Dickmann,F. Miraglia Pdf

In this monograph the authors extend the classical algebraic theory of quadratic forms over fields to diagonal quadratic forms with invertible entries over broad classes of commutative, unitary rings where is not a sum of squares and is invertible. They accomplish this by: (1) Extending the classical notion of matrix isometry of forms to a suitable notion of -isometry, where is a preorder of the given ring, , or . (2) Introducing in this context three axioms expressing simple properties of (value) representation of elements of the ring by quadratic forms, well-known to hold in the field case.

Author : Andor Kertész
Publisher : Unknown
Page : 528 pages
File Size : 51,8 Mb
Release : 1973
Category : Mathematics
ISBN : UCAL:B4407362

Get Book

Rings, Modules and Radicals by Andor Kertész Pdf

Author : Anonim
Publisher : Unknown
Page : 684 pages
File Size : 48,8 Mb
Release : 1979
Category : Aeronautics
ISBN : UIUC:30112075601606

Get Book

Scientific and Technical Aerospace Reports by Anonim Pdf

Lists citations with abstracts for aerospace related reports obtained from world wide sources and announces documents that have recently been entered into the NASA Scientific and Technical Information Database.

Author : Michael Darnel
Publisher : CRC Press
Page : 554 pages
File Size : 44,9 Mb
Release : 2021-12-16
Category : Mathematics
ISBN : 9781000105179

Get Book

Theory of Lattice-Ordered Groups by Michael Darnel Pdf

Provides a thorough discussion of the orderability of a group. The book details the major developments in the theory of lattice-ordered groups, delineating standard approaches to structural and permutation representations. A radically new presentation of the theory of varieties of lattice-ordered groups is offered.;This work is intended for pure and applied mathematicians and algebraists interested in topics such as group, order, number and lattice theory, universal algebra, and representation theory; and upper-level undergraduate and graduate students in these disciplines.;College or university bookstores may order five or more copies at a special student price which is available from Marcel Dekker Inc, upon request.

Author : N. D. Filippov
Publisher : American Mathematical Soc.
Page : 290 pages
File Size : 48,8 Mb
Release : 1970-12-31
Category : Algebra
ISBN : 0821896660

Get Book

Ten papers on algebra functional analysis by N. D. Filippov Pdf

Author : Max Dickmann,Niels Schwartz,Marcus Tressl
Publisher : Cambridge University Press
Page : 652 pages
File Size : 46,7 Mb
Release : 2019-03-21
Category : Mathematics
ISBN : 9781107146723

Get Book

Spectral Spaces by Max Dickmann,Niels Schwartz,Marcus Tressl Pdf

Offers a comprehensive presentation of spectral spaces focussing on their topology and close connections with algebra, ordered structures, and logic.