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As the earlier editions were, this book is intended as a text for an introductory course in algebraic structures (groups, rings, fields, and so forth). Such a course is often used to bridge the gap from manipulative to theoretical mathematics and to help prepare secondary mathematics teachers for their careers. A minimal amount of mathematical maturity is assumed in the text; a major goal is to develop mathematical maturity. The material is presented in a theorem-proof format, with definitions and major results easily located, thanks to a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and providing a bridge to higher-level mathematical courses. Groups appear in the text before rings. The standard topics in elementary group theory are included, and the last two sections in Chapter 4 provide an optional sample of more advanced work in finite abelian groups. The treatment of the set Zn of congruence classes modulo n is a unique and popular feature of this text, in that it threads throughout most of the book. The first contact with Zn is early in Chapter 2, where it appears as a set of equivalence classes. Binary operations of addition and multiplication are defined in Zn at a later point in that chapter. Both the additive and multiplicative structures are drawn upon for examples in Chapters 3 and 4. The development of Zn continues in Chapter 5, where it appears in its familiar context as a ring. This development culminates in Chapter 6 with the final description of Zn as a quotient ring of the integers by the principal ideal (n). Later, in Chapter 8, the use of Zn as a ring over which polynomials are defined, provides some interesting results. Some flexibility is provided by including more material than would normally be taught in one course, and a dependency diagram of the chapters/sections (Figure P.1) is included at the end of this preface. Several sections are marked “optional” and may be skipped by instructors who prefer to spend more time on later topics. Several users of the text have inquired as to what material the authors themselves teach in their courses. Our basic goal in a single course has always been to reach the end of Section 5.3 “The Field of Quotients of an Integral Domain,” omitting the last two sections of Chapter 4 along the way. Other optional sections could also be omitted if class meetings are in short supply. The sections on applications naturally lend themselves well to outside student projects involving additional writing and research. For the most part, the problems in an exercise set are arranged in order of difficulty, with easier problems first, but exceptions to this arrangement occur if it violates logical order. If one problem is needed or useful in another problem, the more basic problem appears first. When teaching from this text, we use a ground rule that any previous result, including prior exercises, may be used in constructing a proof. Whether to adopt this ground rule is, of course, completely optional.