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1999 | Book

Free Modules, Projective, and Injective Modules Read chapter Read first chapter

Lectures on Modules and Rings

Author: T. Y. Lam

Publisher: Springer New York

Book Series : Graduate Texts in Mathematics

Included in: Professional Book Archive

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About this book

Textbook writing must be one of the cruelest of self-inflicted tortures. - Carl Faith Math Reviews 54: 5281 So why didn't I heed the warning of a wise colleague, especially one who is a great expert in the subject of modules and rings? The answer is simple: I did not learn about it until it was too late! My writing project in ring theory started in 1983 after I taught a year-long course in the subject at Berkeley. My original plan was to write up my lectures and publish them as a graduate text in a couple of years. My hopes of carrying out this plan on schedule were, however, quickly dashed as I began to realize how much material was at hand and how little time I had at my disposal. As the years went by, I added further material to my notes, and used them to teach different versions of the course. Eventually, I came to the realization that writing a single volume would not fully accomplish my original goal of giving a comprehensive treatment of basic ring theory. At the suggestion of Ulrike Schmickler-Hirzebruch, then Mathematics Editor of Springer-Verlag, I completed the first part of my project and published the write­ up in 1991 as A First Course in Noncommutative Rings, GTM 131, hereafter referred to as First Course (or simply FC).

Table of Contents

Frontmatter
Chapter 1. Free Modules, Projective, and Injective Modules
Abstract
An effective way to understand the behavior of a ring R is to study the various ways in which R acts on its left and right modules. Thus, the theory of modules can be expected to be an essential chapter in the theory of rings. Classically, modules were used in the study of representation theory (see Chapter 3 in First Course). With the advent of homological methods in the 1950s, the theory of modules has become much broader in scope. Nowadays, this theory is often pursued as an end in itself. Quite a few books have been written on the theory of modules alone.
T. Y. Lam
Chapter 2. Flat Modules and Homological Dimensions
Abstract
This chapter is a natural continuation of Chapter 1 and consists of two long sections. In §4, we study in detail the notion of flat (and faithfully flat) modules, and in §5, we develop the theory of homological dimensions of modules and rings.
T. Y. Lam
Chapter 3. More Theory of Modules
Abstract
In this Chapter, we shall cover some other aspects of the theory of modules that were not yet touched upon in the first two chapters. In contrast to Ch. 1 and Ch. 2, the module theory presented in the three sections of this Chapter is essentially non-homological in nature. Nevertheless, the idea of injective modules and essential extensions plays a discernible role, especially in the first and last sections of the Chapter.
T. Y. Lam
Chapter 4. Rings of Quotients
Abstract
After developing enough module theory in the three previous chapters, the stage is now set for the study of the theory of rings of quotients. The present chapter is a general introduction to this theory, in the setting of noncommutative rings.
T. Y. Lam
Chapter 5. More Rings of Quotients
Abstract
In this Chapter, we study rings of quotients of a different sort, breaking away from the “classical” rings of quotients studied in Chapter 4. Injective modules will play a major role here.
T. Y. Lam
Chapter 6. Frobenius and Quasi-Frobenius Rings
Abstract
The class of rings that are self-injective (as a left or right module over themselves) has been under close scrutiny by ring theorists. There is a vast literature on the structure of self-injective rings satisfying various other conditions. In a book of limited ambition such as this, it would be difficult to do justice to this extensive literature. As a compromise, we focus our attention in this chapter on a special class of such rings called quasi-Frobenius (QF) rings, and the subclass of Frobenius rings. It will be seen that the finite-dimensional Frobenius algebras discussed in §3B are examples of the latter.
T. Y. Lam
Chapter 7. Matrix Rings, Categories of Modules, and Morita Theory
Abstract
This last chapter offers an introduction to the basic categorical aspects of the theory of rings and modules. Since its introduction in the 1940s by Eilenberg and MacLane, the categorical viewpoint has been widely accepted by working mathematicians. For ring theorists especially, the convenient use of the categorical language in dealing with modules serves to provide a unifying force for the subject, and has subsequently become an indispensable tool in its modern study. In this chapter, we shall focus on two of the most important concepts in the application of category theory to rings and modules, namely, the equivalence and duality between two categories of modules. Both of these concepts come from the ground-breaking paper of K. Morita [58], which set in place the basic treatment of these topics pretty much as they are in use today.
T. Y. Lam
Backmatter
Metadata
Title
Lectures on Modules and Rings
Author
T. Y. Lam
Copyright Year
1999
Publisher
Springer New York
Electronic ISBN
978-1-4612-0525-8
Print ISBN
978-1-4612-6802-4
DOI
https://doi.org/10.1007/978-1-4612-0525-8