Undergraduate Algebra: A unified approach
The book under review consists of two parts and the prerequisites. The prerequisites cover primarily some basic Set Theory and can be skipped by a reader already familiar with the notions; however, this section is important to keep the book self-contained.
The two parts are very different by their content, style, and purpose which make this book quite unique. The first part, The Language of Algebra, consists of chapters Glossary of Basic Algebraic Structures, Examples of Groups and Rings, Homomorphisms, and Quotient Structures. The key novelty and unique feature of the book is that, in this first part, analogous topics on different algebraic structures are considered simultaneously (for example, the section on substructures introduces subgroups, subrings, subfields, etc., subsequently; or another example, the section on normal subgroups and quotient groups is followed by the section on ideals and quotient rings). This organization of the book is coherent and surprisingly efficient; it certainly provides a serious alternative to the traditional one where groups, rings, and fields are treated independently. It is interesting to compare the chapters by their styles; for instance, in Chapter 1, the reader can see only some simple examples that may already be familiar, but Chapter 2 consists completely of examples. Ultimately, the main purpose of Part 1 is to provide some basic language of algebraic structures in order to create necessary machinery for Part 2.
The second part, Algebra in Action, consists of the three chapters: Commutative Rings, Finite Groups, and Field Extensions. This is the more dynamic and mathematically appealing part of the book. The presentation is quite different from the first part, as the structure and exposition is closer to how mathematicians truly work. For example, the proof of the Fundamental Theorem of Finitely Generated Torsion Modules resembles the process in which technically involved results are split into smaller claims (in this particular case, seven claims). The book is written in a lively style and is pleasant to read. New concepts are carefully introduced, starting with an informal discussion and, occasionally, historical comments. Each section ends with exercises which progress from easy to challenging, and give a great deal of insight into the subject.
I highly recommend this book for a standard undergraduate algebra course, as well as to students interested in independent study.