国外优秀数学教材选评（第六章 代数）

6代数

数学系本科生课程中，线性代数是最重要的，一般要学一年，其次是抽象代数。一般这些代数知识还不够，在研究生阶段又要学一些更加专门的代数，如群表示、李代数、交换代数、同调代数、环论等。
数学系使用的线性代数教材和非数学专业的教材有很大的不同，前者注重定理的证明，习题中证明题的比重很大，而后者注重计算，习题也以计算题为主，因此非数学专业的学生如果使用数学专业的线性代数教材有可能会不适应。
在此，我们介绍了几本比较基本的优秀代数教材，大多是本科生或低年级研究生使用的。

书名：Linear Algebra, 2nd ed.
作者：K. Hoffmann and R. Kunze
出版商：Prentice Hall, Inc. (1971)
页数：407
适用范围：大学数学系本科低年级教材
预备知识：微积分
习题数量：600 多道习题
习题难度：各种难度都有
推荐强度：9

使用学校：
Central Michigan University，University of North Dakota，Indian Institute of Technology, Bombay，University of Pittsburgh，University of Texas，Johns Hopkins University，West Virginia University，University of Houston，Simon Fraser University，Washington University in St. Louis，University of Notre Dame，University of Wisconsin-Madison，Cornell University，University of South Carolina，University of Rhode Island，University of Missouri，University of Maryland，Stony Brook University，University of Michigan，Purdue University，University of Kansas，United Arab Emirates University，Rice University，Kenyon College，Temple University，Louisiana State University，Sonoma State University，North Carolina State University，University of Iowa

书评： 这是一本线性代数方面久负盛名的教材，在麻省理工学院作为大三课程的教材使用多年，第一版于 1961年出版，对此后世界各国编写的各种线性代数教材有很大影响。比方说，北大和复旦的线性代数教材的内容和编排次序与此书非常接近。
虽然作者在序言中声称为了照顾数学基础不是特别扎实的工科学生把很多地方写的浅一些，但是据我看这本教材只适合数学系或者对线性代数基础要求很高的专业用。
本书内容完整，线性代数的所有内容都有了，甚至多重线性代数的若干内容如外积也包含了。从第一章开始就定下一个基调：本书讲解的线性代数是在任意域上的。在叙述了域的定义后，先给读者一颗定心丸：但是几乎所有例子和习题是在数域上的。接下来马上又警告大家：奇怪的域是
有的，有限多个 1 相加可以等于 0. 这样的安排虽然对学生的要求比较高，我觉得还是利大于弊，可以一步到位了。另一个类似的情形出现在第五章讲行列式时一上来就定义任意交换环上的行列式。我国的教材大多把行列式定义为一个数，然而在例题和习题中很多行列式中含变元，使人觉得不很严格。提起行列式， Strang 教授在 MIT 的视频课程给我深刻印象，他并不先讲行列式的定义，而是先将它需要满足的几个关键性的性质，最后推出只存在唯一的方式来定义行列式满足这些关键性质。本书基本上也是按照这样的方式处理的，虽然费力一些，但把它的本质讲的比较透彻，这比急功近利地急急忙忙讲行列式的计算技巧深刻。更进一步，接下去作者马上讲交换环上模的概念以及交错型和外积，虽然显得有些过火，但这样的安排还是合情合理的，因为这些内容和行列式的关系太密切了。也许作为课堂教学这些内容只能跳过，否则一个学年讲不完这本书。
后半部分的内容是线性变换的各种标准形、内积空间以及它上的线性算子，比较常规。大量习题和这些习题的覆盖度大也是本书的一个特色。（杨劲根）

国外评论摘选
i) I got this book for my Linear Algebra class about four years ago. This is a great book if you are getting a degree in mathematics. It won't help if you are just trying to get by the class and don't like math. It is not very practical but if you are looking for a real math book on Linear Algebra this is it. It contains a wealth of theorems that only a math lover would appreciate. If you really want to learn about Linear Algebra from a rigorous mathematical point of view this is it. This book taught me so much.
ii) This was the textbook they used to use at MIT in the past few decades. Virtually, however, nobody uses this book in a regular undergraduate course anymore. Instead of developing the ideas in the familiar context of the real numbers, Hoffman and Kunze give a more abstract (and general) discussion. For example, the theorems about determinants work in all commutative rings. The rigorousness and the wealth of information are overwhelming for most undergraduates to handle. You will not learn anything if you just glance through the pages. Every line requires deep thought. Down-to-earth applications are not included. So I do not recommend this book for engineers.

书名：Lectures on Linear Algebra
作者：Gelfand
出版商：INTERSCIENCE
PUBLISHERS. INC. （60 年代）
页数：185
适用范围：大学数学系本科低年级教材
预备知识：微积分
习题数量：少
习题难度：比较大
推荐强度：8.5

书评：前苏联有一些数学大师写过一些给大学本科用的数学教材，就象我国华罗庚先生写过《高等数学引论》一样。作为泛函分析祖师爷级别的人物 Gelfand 写的线性代数讲义也不失为一本极好的教材。
本书是学院式的，结构和叙述十分严格和简洁，具有 Bourbaki 的风格，但又不象 Bourbaki 那样地追求一般性，所以不要求读者有很高的起点。不象一般的线性代数教材从解线性方程组或行列式开始，本书从 n 维内积空间作为第一章，甚至包括复内积空间，可谓开门见山。第二章讨论线性变换，特别是正交变换和酉变换。第三章是若当标准型，第四章是多重线性代数。
本书原文是俄文的，在 60 年代有多种文字的译本，包括中文本，由于是繁体字，现在已打入冷宫了。我在大学刚毕业不久（1971年）阅读此书得益非浅，感觉对线性代数乃至泛函分析有更深的认识，这是我把这本老书向读者介绍的主要原因。本人认为此书层次较高，适合于已经有一些初等线性代数知识的人读，它不一定对提高解题能力有利， 但对提高数学观点是绝对有益的。（杨劲根）

国外评论摘选

1) The professor who recommended this book made the comment that every time you re-read it, you notice something else that you missed the last time you read it. This is absolutely true.
I must say, the first time I picked up this book, I did not like it. The notation was not what I was used to, and the book dives right in, assuming a lot of background (matrices, determinants, etc.) but covering material which many people find boring (bases, etc.). However, when you read deeper, there's a lot here. Once you get past the ugly notation, the proofs are extraordinarily clear. And in spite of the books small size, there is a remarkable amount of motivation and discussion.
Like the other reviewer said, this is not a book to learn linear algebra from for the first time: this is an advanced book that is useful for graduate students who have already had a linear algebra course and who want to learn more topics, or understand topics on a deeper level.
This is an excellent book; the bottom line is that it's so cheap that there's no excuse NOT to buy it.

2) This is the best treatment of linear algebra that has been published. It starts with n-dimensional linear spaces and ends with an introduction to tensors. An excellent description of dual spaces is concisely presented. NO INDEX!

3) Lucid and clear notation , complete explanations . This books was first published in 1937 but until now it remains best text book in the field .

4) This is a good book if all you need is a condensed reference on theorems and proofs and it assumes that you go for practice (and instruction) elsewhere. If you are trying to actually learn linear algebra (especially on your own and especially if you want to learn how to solve practical problems) get one of Gilbert Strang's books and watch his video lectures at MIT web site. Another thing that I dislike about the Gelfand's book is that it puts too much emphasis on index notation - instead of matrix notation which is natural for linear algebra, almost all formulas and theorems are presented at very low level using expressions consisting of variables with multiple indices. Naturally it gets very messy and hard to follow at times. This doesn't present any more information than equivalent matrix notation but introduces unnecessary complexity and makes things that are really easy to understand very confusing.

书名：Linear Algebra Gems
作者： David Carlson, Charles R. Johnson,David C. Lay, A. Duane Porter
出版商： The Mathematical Association of America (2000左右)
页数：328
适用范围：大学数学系本科低年级参考读物
预备知识：微积分、线性代数
习题数量：123 题
习题难度：大
推荐强度：8.5

书评： 这本书不是线性代数的教材，而是兴趣浓厚的学生或教线性代数的老师的参考读物，同类 的书并不多见。美国最大的数学组织是美国数学会（AMS）, 其次就是美国数学协会（MAA）, 它 的主要目标是推动数学教学，尤其是大学本科的数学教学，和 AMS 一样，它也有不少出版物， 其中最主要的是美国数学月刊，简称 Monthly, 是一份历史悠久并且享有盛名的数学教育刊物， 上面的文章质量高于我国的数学通报，另有一个刊物 College Mathematics Journal，我国数学界不太熟悉。 本书是从多年的 Monthly 和 College Mathematics Journal 中选出几十篇与线性代数有关 的短文，又约稿请人写了若干文章，总共74篇按内容进行分类而构成的。大部分文章是教学心得和 若干有名的定理（比如若当标准型）和习题的进一步探讨。各篇文章互相独立，每篇文章一般在一个或半个小时内读完， 非常适合于充当大学生课外读物，特别是对大学生数学竞赛很有帮助。
内容共分十部分如下
PART 1 - PARTITIONED MATRIX MULTIPLICATION
PART 2 - DETERMINANTS
PART 3 - EIGENANALYSIS
PART 4 - GEOMETRY
PART 5 - MATRIX FORMS
PART 6 - POLYNOMIALS AND MATRICES
PART 7 - LINEAR SYSTEMS INVERSES AND RANK
PART 8 - APPLICATIONS
PART 9 - OTHER TOPICS
PART 10- PROBLEMS
象第 1，2，5，6，7 部分一看就知道有不少有技巧性的内容。第十部分是习题，大部分是竞赛级别的题。 （杨劲根）

书名：Algebra
作者： Michael Artin
出版商： Prentice Hall (1991), 机械工业出版社影印
页数：618
适用范围：大学数学系本科基础数学一学年的教材
预备知识：微积分和线性代数
习题数量：大
习题难度： 各种难度都有
推荐强度：9.8

书评： 本书是美国大数学家美国科学院院士 Michael Artin 的力作，从70年代早期开始就作为麻省理工学院数学系高年级 本科生教材，是一本极具特色的优秀教材，深受使用者欢迎。 与传统的抽象代数教材不同，本书以数学中的重要实例为主线索，引导出抽象的概念，对读者以启发为主，又不缺乏数学的严格性。虽然教材的主要内容是基本的代数结构，但字里行间不乏现代数学的烙印。代数数论、代数几何、表示论中的一些基本思想 也时时涌现，如整二次型的原理和应用、二次域的理想类、不定方程、紧群表示等。 全书分14章，从矩阵运算引入群概念直到最后一章伽罗华理论一气呵成，不使人感觉600多页篇幅的冗长。 本书的习题是作者20多年积累而得，很多是作者独创的习题，例如有一道2x2魔方的问题是70年代3x3魔方游戏刚问世时作者 编制的群论习题。大约有四分之一的习题有一定难度。 本人80年代在 MIT 攻读研究生期间为此课程作过多次助教，主讲人为作者本人或其他资深教授，每次大约有三十人修课，主要 学生是基础数学各专业的学生，也有一些计算机专业的本科生及研究生选修的。学生反映此课程质量很高，但比较难。
本书比较适合我国综合性大学数学系抽象代数课程的外文教材，尤其适合一学年。对于半年的抽象代数课程，则可选用部分章节。 程度较好的数学系本科生可选用此书作为抽象代数的课外读物。 （杨劲根）

国外评论摘选

1) Pretty much any introductory abstract algebra book on the market does a perfectly competent job of introducing the basic definitions and proving the basic theorems that any math student has to know. Artin's book is no exception, and I find his writing style to be very appropriate for this purpose. What sets this book apart is its treatment of topics beyond the basics--things like matrix groups and group representations. I suppose many introductory books shy away from much of the material on matrix groups in Artin's book because it involves a little analysis (and likewise for the section on Riemann surfaces in the chapter on field theory). However, Artin correctly realizes that a reasonably mathematically mature student--even one who doesn't know much analysis--will be able to profit from and enjoy the relatively informal treatments he gives these slightly more advanced topics. Of course these topics can also be found in graduate-level texts, but I for one would much rather be introduced to them via an example-based approach such as that in Artin than through the diagram-chasing obscurantism in more advanced books. I happened upon this book a little late--in fact, only after I'd taken a semester of graduate-level algebra and already felt like analysis was the path I wanted to take--but I'm beginning to think I would have been more keen on going into algebra if I'd first learned it from a book like this one.
2) I bought this book for a class that I ended up dropping. In the beginning, I hated this book. I found Herstein's topics in algebra much better, and more to the point. It was only when I was getting bored with Herstein that I bothered to pick this up again. I was pleasantly surprised. A lot of the material flowed very smoothly - exactly as if Artin was teaching the material to you. It must however be noted that people tend to love or hate this book. This is predominantly due to the author's writing style. Given how expensive this book is, you might perhaps want to peruse it somewhere before deciding to buy it. But if you do, you'll get a solid exposition on most of the introductory topics in algebra as well as some insight on groups and symmetry, lie groups, representation theory, galois theory and quadratic number fields. And a whole lot of intuition as well, for the more regular topics. Give this book a chance - it's worth the effort and money.
3) As an undergraduate I learned, or tried to learn, algebra from this book. Artin's pedagogical methods just didn't work for me. Although his idea of teaching through concrete, geometric examples sounds great in principle, in practice it's not so successful. It is very hard to see the forest for the trees, since Artin is so chatty and discursive. When he is discussing examples, he sometimes puts specialized results on par with more general theorems, which may be misleading. Many proofs are only sketched, and occasionally theorems are stated after their proofs, necessitating a rereading of the preceding paragraphs in order to grasp the points of the proof. The chapters on representation theory (Ch. 9) and arithmetic of quadratic number fields (Ch. 11) are nonstandard topics and interesting in themselves, but again, the level of detail tends to obscure, rather than enlighten.
The one saving grace of the book is the excellent problem sets at the end of each chapter. In doing them you will learn the algebra that the main body of the text attempts to impart.

书名：Codes and Curves
作者： Judy Walker
出版商： American Mathematical Society (2000)
页数：66
适用范围：大学数学系本科高年级参考书
预备知识：抽象代数
习题数量：小
习题难度： 容易
推荐强度：8.5

书评： 这本小册子是1999年美国数学会在 Princeton 组织的暑期学校的一门课程的讲稿，是代数几何码的入门读物。代数几何码是新发现的 一种纠错码，目前仍有大量问题在研究。本书前一半对纠错码的基本知识和若干经典的纠错码作了扼要的介绍，重点是 Reed-Solomon 码，因为代数几何码是它的推广。然后，作者不加证明地清楚地叙述了有限域上平面代数曲线的基本知识， 最后介绍了代数几何码以及好的代数几何码的构造方法。
本书的一个显著特点是提供了六个供本科生研究的课题。本人曾指导复旦大学数学系毕业班的六名学生报告这本书，并围绕 六个课题查阅文献资料，写作毕业论文，取得很好的效果。 （杨劲根）

国外评论摘选

1) The book gives an overview of algebraic coding theory. The first chapter introduces error correcting codes, the Hamming distance, Reed-Solomon codes, and concludes with a brief exposition of cyclic codes. The second chapter discusses some upper bounds on the minimum distance of a code such as the Singleton and Plotkin bounds.
The second theme of this book are algebraic curves. Chapter 3 contains the basic definitions and some examples of algebraic curves. The concept of a nonsingular curve is explained in Chapter 4. This chapter also contains a half page explanation of the genus of a curve. The Riemann-Roch theorem is finally covered in Chapter 5.
The two themes come together in Chapters 6 and 7. These chapters discuss the basic principles of algebraic geometry codes.
This little book gives the reader a first taste of an intriguing field. The most surprising part is how much is covered in so few pages [the main text without appendices has 44 pages]. The explanations are always accessible for undergraduate students of mathematics, computer science, or electrical engineering. The prerequisites are some knowledge of abstract algebra, but most material is reviewed in the appendices.
It is a lovely little book that is written in a lively style. The book nicely complements the typical college courses on coding theory. If you want to get an idea what algebraic geometric codes are and you want a quick answer, then this is the book for you.
2) There is a free version of the book available on the website of the University of Nebraska-Lincoln.
目录：
Chapter 1. Introduction to Coding Theory
1.1. Overview
1.2. Cyclic Codes
Chapter 2. Bounds on Codes
2.1. Bounds
2.2. Asymptotic Bounds
Chapter 3. Algebraic Curves
3.1. Algebraically Closed Fields
3.2. Curves and the Projective Plane
Chapter 4. Nonsingularity and the Genus
4.1. Nonsingularity
4.2. Genus
Chapter 5. Points, Functions, and Divisors on Curves
Chapter 6. Algebraic Geometry Codes Chapter
7. Good Codes from Algebraic Geometry
Appendix A. Abstract Algebra Review
A.1. Groups
A.2. Rings, Fields, Ideals, and Factor Rings
A.3. Vector Spaces
A.4. Homomorphisms and Isomorphisms
Appendix B. Finite Fields
B.l. Background and Terminology
B.2. Classification of Finite Fields
B.3. Optional Exercises
Appendix C. Projects
C.1. Dual Codes and Parity Check Matrices
C.2. BCH Codes
C.3. Hamming Codes
C.4. Golay Codes
C.5. MDS Codes
C.6. Nonlinear Codes

书名：Introduction to Commutative Algebra
作者： Michael Atiyah & I.G.MacDonald
出版商： Addison-Wesley Publishing Company (1991)
页数：126
适用范围：大学数学系本科基础数学高年级或研究生低年级教材
预备知识：抽象代数和点集拓扑
习题数量：大
习题难度： 较大
推荐强度：9

书评： 英国皇家科学院院士 Michael Atiyah 是当代大数学家，曾或菲尔滋奖。本书是交换代数的入门书籍，是一本优秀教材， 特别适合于代数几何、代数数论和其他代数专业的研究生使用。本书的篇幅虽小，内容却很丰富，包含了交换代数的核心内容。 学过一学期抽象代数的人可以顺利学习本书前九章，学习第十和第十一章需要点集拓扑的基本知识。正文中的定理的证明简明易懂，有很多重要的定理安排在习题中，所以要掌握此书内容必须化工夫做每一章后的大部分习题。
本书以诺特交换环和有限生成模作为重点，这正是代数几何和代数数论中出现最多的代数结构。 作者在序言中说到域论没有 涉及，这可以从别的优秀教材（如 Nagata 的“域论”）中得到补充。
国外很多名校的数学教授将此书作为交换代数教材的首选。我国引进此书也很早，它很受师生的欢迎。 （杨劲根）

国外评论摘选

1) Some people believe that, for getting into algebraic geometry (by this I mean Grothendieck-like AG, with schemes and all that), one needs a monolithic training in commutative algebra (something like both volumes of Zariski-Samuel, for example). I disagree. This little book seems to be specially suited to those who want to learn AG. It's a bit too brisk, specially at the beginning - if you don't already have an acquaintance with the basics of groups, rings and ideals, you may run into trouble - but very illuminating. Masterful choice of topics, great exercises (as a matter of fact, about half the topics of the book, and more specifically the ones that are directly related to AG, are treated in the exercises, some of them quite challenging) - like one said before, it looks like a chapter 0 of Hartshorne's book on AG. The authors consciously estabilish relations between the commutative algebra and the modern foundations of AG over and over along the way, illuminating both topics. For the algebra itself, it also gets on well with Rotman's Galois Theory and MacDonald's out-of-print introduction to AG, Algebraic Geometry - Introduction to Schemes, besides being the perfect preamble in commutative algebra to the books of Mumford and Hartshorne. A gem.
2)The strongest aspects of Atiyah & MacDonald's book are its brevity, accessibility to undergraduates, and subtle introduction of more advanced material.
Audience: I think an undergraduate with a solid understanding of material from a first course in abstract algebra (i.e., the chapter on rings--the modules chapter would help, but isn't necessary--from M. Artin's book 'Algebra' is more than sufficient) and some basic point-set topology from an intro real analysis course (or ch1-4 of Munkres) would be sufficient for fully appreciating the material. I think having experience in PS Topology is important for understanding parts of this book well; doing the exercises is possible if you learn it on the fly, but I hadn't seen Urysohn's Lemma before, and even that caused me some intuition hangups; to fully appreciate the material, I would recommend doing a healthy number of problems in topology first.
Material: The material uses concepts from homological algebra, though in a disguised form; students with experience in category theory will find offhanded comments that recast some of the material in that language, but CT is absolutely not essential to understand the material well. It also provides exercises that lead naturally into topics from Algebraic Geometry and Algebraic Number Theory quite readily; a nice set of problems in CH1 walk a student through construction of the Zariski topology, prime spectrum, etc., and some functional properties of morphisms between spectra. Algebraic Number Theory starts showing up after chapter 4 in greater detail, and would lead comfortably into Lang's GTM on ALNT by CH9 (though I only read a bit of Lang, the first chapter felt natural).
The details left to the reader are usually reasonably tackled with the tools made available so far, and the book is short enough that one can cover a lot of ideas in a reasonable amount of time; the commentary made by the authors is brief, to the point, and never redundant as far as I can recall, so I consider this a highly efficient book (but not too efficient, it's self contained enough and not uncompromisingly terse).
Exercises: They are quite good, I think. Very few of them follow from symbol-pushing or robotic theorem proving, and usually require some constructive argument. The exercises are mostly chosen to introduce more advanced material, and do a good job in that regard. The longer chapters have 25-30 exercises, and shorter chapters (a few pages) have maybe 10, so there are plenty of problems to do.
Hazards: The material on modules is brisk, the propositions in the first three sections on modules are mostly left without proof; however, the proofs follow from their analogues for rings, and aren't that hard, just be sure to actually do them because they are mentioned only briefly. Also, the book is not typo-free, but this only caused me one major hangup during the semester. After Chapter 3, the proofs are mostly complete, with a spattering of left to the reader exercises, which I usually found helpful.
Companion Material: I think Lang's 'Algebra' GTM would make a nice reference for the material on Homological Algebra and other miscellaneous things that come up in the proofs; I remember once a proof in the book required the notion of the adjoint of a matrix over a ring, and so I had to look it up in Lang, and also the basic category theory covered in CH1 of Lang would at least introduce (though in a very rapid way) the abstract nonsense mentioned offhandedly here and there. If you have a lot of money, or access to a good library, 'Categories for the Working Mathematician' is a slower and more thorough introduction to that language, and I would recommend at least having a look, though this isn't really central to the material from Commutative Algebra.
3) This is how mathematics texts SHOULD be written. As in technical writing, the smaller text is the better written text. Everything is clean and direct, with clairity obviously a prime consideration. One never gets mired down. The proofs are always as close to a THE BOOK proof as possible, with illuminating examples, and plenty of excercises, many with outlines for solution, which makes the book ideal for self study. This book is a revelation. If I had to take only one math text with me to a desert island, this would be the one.
4) This is a difficult book for undergraduates, even ones who have already had some abstract algebra. Many refer to the book's style as terse, meaning that there is little explanation, few examples, and proofs are very condensed.

书名：HOPF ALGEBRAS
作者： MOSS E. SWEEDLER
出版商： W. A. BENJAMIN, INC. (1969)
页数：336
适用范围：大学数学系本科、数学专业研究生
预备知识：代数、环模基础理论
习题数量：小
习题难度： 容易
推荐强度：9

书评： 1941年，德国数学家H. Hopf在研究代数拓扑时引入了Hopf代数的概念。真正引 起人们对这类代数结构普遍关注的是1965年J.W. Milnor 和J.C. Moore的有关分 次Hopf代数的文章；到上世纪80年代末，量子群概念的出现及其在Knot不变量理 论中的应用将Hopf代数的研究推向一个新的高潮。如今，Hopf代数理论正在诸如代 数群、李代数、表示论、组合论以及量子力学等学科的研究中发挥着重要的作用。M.E. Sweedler所著的\textquotedblleft Hopf Algebra\textquotedblright是历史上第 一本系统介绍这方面Hopf代数知识的书籍。
这本书是从Sweedler给研究生的系列讲座内容中整理出来的，介绍的对象主要是非分 次的Hopf代数。域$k$上一个增广（augmented）代数$H$，若带有一个余结合的 （coassociative）和余单位的（counitary）代数映射$\Delta \colon H\rightarrow H\otimes H$，则称$H$是一个双代数（bialgebra），Hopf代数是指带 有antipode的双代数。本书开始先引入sigma-符号；而后一步步将上世纪70年代 以来有关Hopf代数的最新结果，其中大部分是作者与其合作者当时取得的进展，呈现 给读者；最后一章以证明域上由所有有限维交换、余交换的Hopf代数组成的范畴是交 换的（abelian）范畴作为结束。整本书的内容简洁，易懂，且自我包含，是一部很好 的关于Hopf代数知识的入门教材。它的不尽完美之处是没有列出任何参考文献 。
本书共有16章，有些章节非常短。前4章，给出了余代数、余模、及Hopf代数的 初步介绍，其中包括从模中构建余模的有理模构造方法及余代数的基本定理---该定 理阐明了在一个余代数中，任意有限个元素均包含于一个有限维子余代数中，故而任一 余代数都是有限维余代数的直接极限。第5章讨论了积分（integral）， \textquotedblleft积分\textquotedblright这一名称是因它很像紧群上关于 Haar测度的积分运算而得名。这一章的重要结果是证明任一有限维Hopf代数必存在 一维的积分空间；作为一个推论，本章中导出了群环的著名Maschke定理。
第6章对一个代数引入并讨论了它的对偶余代数。第7章到第9章主要介绍了度量 、smash积和外积等概念，为第13章余交换点（pointed）Hopf代数的结构定理的 证明作了前期的准备工作。
第10章的主要内容是Hopf代数作用的Galois理论；第11章对本原元进行了重点 讨论并引入了分次双代数的概念，在这一章以及后面的第14章中余代数的基本定理起 到了很大的作用。第12章考虑了shuffle代数和相关既约的点余代数万有映射性质 ，此外还讨论了divided power。
第13章中证明了著名的结论：任一既约的点余交换Hopf代数一定同构于Lie代数 或限制Lie代数的包络代数；据此导出了余交换Hopf代数的结构定理。剩下的两章 讨论了仿射群和由交换、余交换Hopf代数构成的Abelian范畴。 （朱胜林）