Algebra (Cambridge Mathematical Textbooks)

Book Preface

This is an introductory textbook on abstract algebra. A glance at the table of contents will immediately reveal its basic organization and salient features: it belongs to the ‘rings-first’ camp, and places an unusual emphasis on modules, which (to my puzzlement) are often essentially omitted in textbooks at this level. A few section titles will also reveal passing nods to the notion of category, which is not developed or used in the main body of the text, but is also not intentionally hidden from sight.

Why ‘rings first’? Why not ‘groups’? This textbook is meant as a first approach to the subject of algebra, for an audience whose background does not include previous exposure to the subject, or even very extensive exposure to abstract mathematics. It is my belief that such an audience will find rings an easier concept to absorb than groups.
The main reason is that rings are defined by a rich pool of axioms with which readers are already essentially familiar from elementary algebra; the axioms defining a group are fewer, and they require a higher level of abstraction to be appreciated. While Z is a fine example of a group, in order to view it as a group rather than as a ring, the reader needs to forget the existence of one operation. This is in itself an exercise in abstraction, and it seems best to not subject a na¨ıve audience to it. I believe that the natural port of entry into algebra is the reader’s familiarity with Z, and this familiarity leads naturally to the notion of ring. Natural examples leading to group theory could be the symmetric or the dihedral groups; but these are not nearly as familiar (if at all) to a na¨ıve audience, so again it seems best to wait until the audience has bought into the whole concept of ‘abstract mathematics’ before presenting them.

Thus, my choice to wait until later chapters before introducing groups is essentially dictated by the wish to provide a gentle introduction to the subject, where the transition to the needed level of abstraction can happen gradually. The treatment in the first several sections of this book is intentionally very elementary, with detailed explanations of comparatively simple material. A reader with no previous exposure to this material should be able to read the first two chapters without excessive difficulty, and in the process acquire the familiarity needed to approach later chapters. (Rings are introduced in Chapter 3.) The writing becomes more demanding as the material is developed, as is necessary—after all, the intended readership is expected to reach, by the end of the book, a good comfort level with rather sophisticated material, such as the basics of Galois theory. It is my hope that the reader will emerge from the book well equipped to approach graduate-level algebra. In fact, this is what many students in my undergraduate algebra courses have done.

The book could be used for a group-first approach, provided that the audience is ‘mature’ enough to cope right away with the higher level of abstraction of groups, and that the instructor is willing to occasionally interpolate references to previous chapters. In fact, the reliance of the chapters on groups in Part III on material developed in previous chapters is minimal, and mostly contextual (with the exception of a treatment of cyclic groups, which is carried out in Part II). However, the writing in Part III is naturally terser and puts more demands on the reader; so, again, this is not recommended for a na¨ıve audience. Part IV deals with fields, and here the audience is certainly expected to be comfortable handling abstract concepts, perhaps at the level of a Master’s student. I would definitely not recommend a ‘fields-first’ approach in an introduction to algebra.

More than 50 years ago, Atiyah and Macdonald wrote in their Introduction to Commutative Algebra: “. . . following the modern trend, we put more emphasis on modules and localization.” Apparently, what was the ‘modern trend’ in 1970 has not really percolated yet to the standard teaching of an introductory course in algebra in 2020, since it seems that most standard textbooks at this level omit the subject of modules altogether. In this book, modules are presented as equal partners with other standard topics—groups, rings, fields. After all, most readers will have had some exposure to linear algebra, and therefore will be familiar to some extent with vector spaces; and
they will emerge from studying rings with a good understanding of ideals and quotient rings. These are all good examples of modules, and the opportunity to make the reader familiar with this more encompassing notion should not be missed.
The opportunity to emphasize modules was in itself sufficient motivation to write this book. I believe that readers will be more likely to encounter modules, complexes, exact sequences, introduced here in §9.2, in their future studies than many other seemingly more standard traditional topics covered in introductory algebra books (and also covered in this text, of course).

Having made this choice, it is natural to view abelian groups as particular cases of modules, and the classification theorem for finitely generated abelian groups is given as a particular case of the analogous results for finitely generated modules over Euclidean domains (proved in full). Groups are presented as generalizations of abelian groups; this goes counter to the prevailing habit of presenting abelian groups as particular cases of groups. After treating groups, the text covers the standard basic material on field extensions, including a proof of the Fundamental Theorem of Galois Theory. On the whole, the progression rings–modules–abelian groups–groups–fields simply seems to me the most natural in a first approach to algebra directed at an audience without previous exposure to the subject. While it is possible to follow a different progression (e.g., start with groups and/or avoid modules) I would have to demand more of my audience in order to carry that out successfully.

I could defend the same conclusion concerning categories. My graduate(-leaning) algebra textbook1 has somehow acquired the reputation of ‘using categories’. It would be more accurate to state that I did not go out of my way to avoid using the basic language 1 Algebra: Chapter 0. Graduate Studies in Mathematics, No. 104. American Mathematical Society, Providence, RI. of categories in that textbook. For this textbook, aimed at a more na¨ıve audience, I have resolved to steer away from direct use of the language. I have even refrained from giving the definition of a category! On the other hand, the material itself practically demands a few mentions of the concept—again, if one does not go out of one’s way to avoid such mentions. It is possible, but harder, to do without them. It seemed particularly inevitable to point out that many of the constructions examined in the book satisfy universal properties, and in the exercises the reader is encouraged to flesh out this observation for some standard examples, such as kernels or (co)products. My hope is that the reader will emerge from reading these notes with a natural predisposition for absorbing the more sophisticated language of categories whenever the opportunity arises.

The main prerequisite for this text is a general level of familiarity with the basic language of naive set theory. For the convenience of the reader, a quick summary of the relevant concepts and notions is included in the Appendix A, with the emphasis on equivalence relations and partitions. I describe in some detail a decomposition of set-functions that provides a template for the various ‘isomorphism theorems’ encountered in the book. However, this appendix is absolutely not required in order to read the book; it is only provided for the convenience of the reader who may be somewhat rusty on standard set-theoretic operations and logic, and who may benefit from being reminded of the various typical ‘methods of proof’. There are no references to the appendix from the main body of the text, other than a suggestion to review (if necessary) the all-important notions of equivalence relations and partitions. As stressed above, the text begins very gently, and I believe that the motivated reader with a minimum of background can understand it without difficulty.

It is also advisable that readers have been exposed to a little linear algebra, particularly to benefit the most from examples involving, e.g., matrices. This is not strictly necessary; linear algebra is not used in any substantial way in the development of the material. The linear algebra needed in Part II is developed from scratch. In fact, one subproduct of covering modules in some detail in this text should be that readers will be better equipped to understand linear algebra more thoroughly in their future encounters with that subject.

Appendix B includes extensive solutions to about one-third of the exercises listed at the end of each chapter; these solved problems are marked with the symbol . These are all (and only) the problems quoted directly from the text. In the text, the reader may be asked to provide details for parts of a proof, or construct an example, or verify a claim made in a remark, and so on. Readers should take the time to perform these activities on their own, and the appendix will give an opportunity to compare their work with my own ‘solution’. I believe this can be very useful, particularly to readers who may not have easy access to an alternative source to discuss the material developed in this book.
In any case, the appendix provides a sizeable amount of material that complements the main text, and which instructors may choose to cover or not cover, or assign as extra reading (after students have attempted to work it out on their own, of course!).

How to use this book? I cover the material in this book in the order it is written, in a two-semester sequence in abstract algebra at the undergraduate level at Florida State University. Instructors who, like me, have the luxury of being able to spend two semesters on this material are advised to do the same. One semester will suffice for the first part, on rings; and some time will be left to begin discussing modules. The second semester will complete modules and cover groups and fields.

Of course, other ways to navigate the content of this book are possible. By design, the material on groups has no hard references back to Chapter 6, 7, or 9. Therefore, if only one semester is available, one could plan on covering the core material on rings (Chapters 1–5), modules (Chapter 8), and groups (Chapter 11). It will be necessary to also borrow some material from Chapter 10, such as the definition and basic features of cyclic groups, as these are referenced within Chapter 11. This roadmap should leave enough time to cover parts of Chapter 12 (more advanced material on groups) or Chapter 13 (basic material on field extensions), at the discretion of the instructor.

The pedagogical advantages of the rings-first approach have been championed in Hungerford’s excellent undergraduate-level algebra text, which I have successfully used several times for my courses. Readers who are familiar with Hungerford’s book will detect an imprint of it in this book, particularly in the first several sections and in the choice of many exercises.

Thanks are due to the students taking my courses, for feedback as I was finding this particular way of telling this particular story. I also thank Ettore Aldrovandi for spotting a number of large and small errors in an earlier version of these notes, and several anonymous reviewers for very constructive comments. I thank Sarah Lewis for the excellent copyediting work, and Kaitlin Leach and Amy Mower at Cambridge University Press for expertly guiding the book from acquisition through production. Lastly, thanks are due to the University of Toronto and to Caltech, whose hospitality was instrumental in bringing this text into the present form