**Graduate Studies in Mathematics**

Volume: 104;
2009;
713 pp;
Softcover

MSC: Primary 00;
Secondary 12; 13; 15; 18; 20

**Print ISBN: 978-1-4704-6571-1
Product Code: GSM/104.S**

List Price: $95.00

AMS Member Price: $76.00

MAA Member Price: $85.50

**Electronic ISBN: 978-1-4704-1168-8
Product Code: GSM/104.E**

List Price: $89.00

AMS Member Price: $71.20

MAA Member Price: $80.10

#### You may also like

#### Supplemental Materials

# Algebra: Chapter 0

Share this page
*Paolo Aluffi*

Algebra: Chapter 0 is a self-contained introduction to the main topics of algebra, suitable for a first sequence on the subject at the beginning graduate or upper undergraduate level. The primary distinguishing feature of the book, compared to standard textbooks in algebra, is the early introduction of categories, used as a unifying theme in the presentation of the main topics. A second feature consists of an emphasis on homological algebra: basic notions on complexes are presented as soon as modules have been introduced, and an extensive last chapter on homological algebra can form the basis for a follow-up introductory course on the subject. Approximately 1,000 exercises both provide adequate practice to consolidate the understanding of the main body of the text and offer the opportunity to explore many other topics, including applications to number theory and algebraic geometry. This will allow instructors to adapt the textbook to their specific choice of topics and provide the independent reader with a richer exposure to algebra. Many exercises include substantial hints, and navigation of the topics is facilitated by an extensive index and by hundreds of cross-references.

#### Readership

Undergraduate and graduate students interested in algebra.

#### Reviews & Endorsements

An obvious question: why another graduate algebra book? Aren't there all but too many already? Is there anything genuinely novel to be done when it comes to educating fledgling graduate students in this subject, given such an already well-populated and high-quality field? Well, the answer is yes. And the title of the book under review, Algebra: Chapter 0, is already a clue to what the author, Paolo Aluffi, is up to. In a perhaps Bourbakian sense, the prevailing motivation and objective is to present the subject at hand in a manner that pays proper due to relatively new foundations, making for a rather different orientation and flavor for what ensues. ...His treatment of these preliminaries is thorough as well as eminently accessible: Aluffi writes well, clearly and engagingly. This characterizes all of Algebra: Chapter 0, actually, and makes it easy to recommend the book enthusiastically even aside from the fact that I am a big fan of category theory to begin with.

-- MAA Online

This self-contained introduction is suitable for a first sequence at the beginning graduate or upper undergraduate level. A distinguishing feature of the book is the early introduction of categories, used as a unifying theme.

-- SciTech Book News

#### Table of Contents

# Table of Contents

## Algebra: Chapter 0

Table of Contents pages: 1 2

- Title page 44
- Contents 66
- Preface to the second printing 1616
- Introduction 1818
- Chapter I. Preliminaries: Set theory and categories 2424
- 1. Naive set theory 2424
- Exercises 3131
- 2. Functions between sets 3131
- 2.1. Definition 3131
- 2.2. Examples: Multisets, indexed sets 3333
- 2.3. Composition of functions 3333
- 2.4. Injections, surjections, bijections 3434
- 2.5. Injections, surjections, bijections: Second viewpoint 3535
- 2.6. Monomorphisms and epimorphisms 3737
- 2.7. Basic examples 3838
- 2.8. Canonical decomposition 3838
- 2.9. Clarification 3939

- Exercises 4040
- 3. Categories 4141
- Exercises 4949
- 4. Morphisms 5050
- Exercises 5353
- 5. Universal properties 5454
- Exercises 6161

- Chapter II. Groups, first encounter 6464
- 1. Definition of group 6464
- Exercises 7171
- 2. Examples of groups 7272
- Exercises 7979
- 3. The category \Grps 8181
- Exercises 8686
- 4. Group homomorphisms 8787
- Exercises 9292
- 5. Free groups 9393
- Exercises 101101
- 6. Subgroups 102102
- Exercises 108108
- 7. Quotient groups 111111
- Exercises 118118
- 8. Canonical decomposition and Lagrange’s theorem 119119
- Exercises 128128
- 9. Group actions 131131
- Exercises 136136
- 10. Group objects in categories 138138
- Exercises 140140

- Chapter III. Rings and modules 142142
- 1. Definition of ring 142142
- Exercises 150150
- 2. The category \Rings 152152
- Exercises 159159
- 3. Ideals and quotient rings 161161
- Exercises 166166
- 4. Ideals and quotients: Remarks and examples. Prime and maximal ideals 167167
- Exercises 176176
- 5. Modules over a ring 179179
- Exercises 186186
- 6. Products, coproducts, etc., in \Rmod 187187
- Exercises 195195
- 7. Complexes and homology 197197
- Exercises 206206

- Chapter IV. Groups, second encounter 210210
- Chapter V. Irreducibility and factorization in integral domains 266266

Table of Contents pages: 1 2