1.1. Review by: B H Neuman.
The Mathematical Gazette 44 (348) (1960), 150-151.

It takes courage to write a treatise on abelian groups so soon after the appearance of Kaplansky's lively little book. It takes courage also to write such a book in a language not one's own. The book under review is a testimony not only to the author's great courage, but also to his tremendous - and infectious - enthusiasm. There are small blemishes, a minor misprint here, a slight idiomatic oddity there, but all this is insignificant compared to the many excellent features of the book. Professor Fuchs has packed a very great amount of important and interesting group theory into it; he always explains what he intends to do, and why; he not only proves theorems but also discusses their significance. The vast amount of material selected for presentation includes everything that is of real importance; many of the lesser results are relegated to the exercises that accompany every chapter. There are some 550 illustrative exercises in all, many with hints for solution, others with asterisks to warn of special difficulty. Many unsolved problems are formulated. Numerous careful references, a comprehensive bibliography, indices of authors, subjects, and notation contribute to make the book as useful a work of reference for the initiate as it is an inspiring introduction for the novice.

1.2. Review by: F W Levi.
Mathematical Reviews MR0106942 (21 #5672).

The title of this book marks its content as well as the point of view from which the author looks at the subject. "Abelian groups" are considered as an independent part of mathematics, not as a special case of group-theory, nor are groups regarded as a special case of "groups with operators". The huge amount of results presented in this work - much of it due to the Hungarian school of mathematicians and in particular to the author - seems to justify this limitation, which provides the book with a proper style and even with its own grammar. "Group" is always used in the sense of Abelian group, the group composition is denoted by + and a consistent system of notations is applied throughout the book. A table of notations at the beginning and a subject index at the end give sufficient information about the notations even to those readers who are not reading the book from cover to cover. Whereas the author avoids all rhetoric, he does not spare words for giving information about the interconnection of the various problems. The proofs are given in full; sometimes several essentially different proofs are given for important theorems. It is a very readable book. Each chapter concludes with a large list of exercises - altogether more than five hundred - and some unsolved problems. In the preface, the author has expressed the hope that these problems will help to promote the research on Abelian groups. Indeed some of them have been solved in the meantime. At the end of this review, a list of solutions will be given.

1.3. Review by: R A Beaumont.
Bulletin of the American Mathematical Society 66 (1960), 480-482.

This encyclopaedic work on the theory of abelian groups will certainly further stimulate the interest in this subject which has been growing steadily since the publication of Part II of Kurosh's Group theory (1953) and Kaplansky's Infinite abelian groups (1954). This is the first book which includes, along with the fundamental structure theory which appears in the earlier mentioned works, the large body of material contributed in the last several years by the Hungarian school of group theorists. The material in Kurosh is entirely covered. The generalisations to modules over a principal ideal domain and a complete discrete valuation ring, as well as the applications to topological groups, which are features of Kaplansky's monograph, are not included except in the exercises. The author believes that a module-theoretic treatment would imply either almost trivial generalisations and unnecessary complications in the discussions, or deeper extensions which are of a ring-theoretic rather than a group-theoretic nature. The author has been eminently successful in giving a complete, detailed, and easily understandable account of the present status of the theory of abelian groups with special emphasis on results concerning structure problems.
...
Among many notable features of this book which should be mentioned are the excellent bibliography, the exercises with a wide range of difficulty which cover virtually every topic presented, and the statement of eighty-six unsolved problems which already have led to new contributions to the theory of abelian groups. The book is printed in the same large clear type and format as the Hungarian mathematical journals and is remarkably free of misprints.

This book is an important addition to mathematical literature and is highly recommended to anyone whose interests touch the theory of abelian groups.

1.4. Review by: E H Batho.
The American Mathematical Monthly 67 (8)(1960), 816-817.

The combination in one book of treatise and textbook is not a new phenomena in mathematical publishing. However, rarely has it been done with such felicity and success as in Fuchs' book. Starting with the fundamental ideas, the book carries the beginning student through all the principal areas such as cyclic groups, divisible groups, pure subgroups, torsion and torsion-free groups, etc. At the same time and particularly in the latter half of the book - the specialist will find an almost exhaustive treatment of current research in Abelian groups, e.g. the structure of additive groups of rings and multiplicative groups of fields.

The treatment throughout is detailed and extremely clear. The subject matter is so comprehensive that it might be advisable for teachers to point out to a beginning student those parts that might best be omitted on a first reading lest he be lost in too much detail. One of the happiest features of the book is the abundance of exercises and problems at the end of each chapter. The exercises are never trivial and range from the fairly direct to the quite complicated. The problems (86 in all) are all problems which were unsolved as of the writing of this book. This book should do a great deal to stimulate interest in these problems. A thorough bibliography is given. Finally, the book seems to be free from any major misprints.

A listing of the subject matter covered is as follows: Basic concepts; Direct sums of cyclic groups; Divisible groups; Direct summands and pure subgroups; Basic subgroups; The structure of p-groups; Torsion-free groups; Mixed groups; Homomorphism groups and Endomorphism rings; Group extensions; Tensor products; The additive group of rings; The multiplicative group of fields; The lattice of subgroups; Decompositions into direct sums of subsets; Miscellaneous questions.

2.1. From the Preface.

In recent years, interest in the study of partially ordered groups, semigroups, rings and fields has been increasing. The many results in numerous papers are widely spread over the journals, and no systematic survey exists. With the present book I am trying to fill this gap, and to present the most essential results to those who want to become acquainted with this subject.

Algebraic systems endowed with a partial or full order are met with in several disciplines of mathematics. Because the theory of partially ordered algebraic systems is extensive, I have had to abandon the claim to completeness and to be content with developing the main algebraic aspects of the theory. This is the reason why some important topics, such as partially ordered linear or topological spaces, have not been taken into consideration. Moreover, a certain limitation was also necessary in the presentation of results which are purely algebraic in character. To enable the reader to find out more about the subject, references have been provided not only to the original sources of the material here collected, but also to a number of important results which I have not been able to include. No attempt has been made to cover the whole vast field of partially ordered algebraic systems in the bibliography, but for the narrower field surveyed here, the bibliography should be fairly complete.

The text falls into three main parts. I chose the theory of partially ordered groups for the first part, because it is more important both conceptually and from the point of view of the general theory, than the theory of partially ordered semigroups. The second part is devoted to the exposition of partially ordered rings and fields, while the third is concerned with partially ordered semigroups. Some attention has been paid to the non-associative case as well. In order to underline the intrinsic analogies, I endeavoured to set up the three parts on parallel lines as far as possible - this is apparent also from the titles of the chapters.

I have tried to make the presentation self-contained and to give complete proofs of the results. However, in some places it was inevitable to assume some previous knowledge of more or less known results of abstract algebra. In these cases either the needed background is reproduced or references are given. The most familiar concepts of algebra are used, of course, without comment.

2.2. Contents of Partially ordered algebraic systems.

Preface

Table of notations

CHAPTER I. Introduction
1. Partially ordered sets
2. Partial order in algebraic systems

FIRST PART: PARTIALLY ORDERED GROUPS

CHAPTER II. Preliminaries on partially ordered groups
1. Definitions
2. The positive cone
3. Examples
4. Subgroups and factor groups
5. 0-homomorphisms
6. Direct products
7. Lexicographic products
8. Intrinsic topologies

CHAPTER III. Extensions of partial orders in groups
1. Extension to a full order
2. O-groups
3. Some group-theoretical properties of O-groups
4. O*-groups
5. Intersection of full orders
6. Vector groups

CHAPTER IV. Fully ordered groups
1. Archimedean fully ordered groups
2. Full orders on free groups
3. The chain of convex subgroups
4. Valuations of fully ordered Abelian groups
5. Hahn's embedding theorem
6. Cyclically ordered groups

CHAPTER V. Lattice-ordered groups
1. Algebraic rules
2. Orthogonality
3. Carriers
4. Positive and negative parts; absolutes.
5. l-ideals
6. Groups with a finite number of carriers
7. Units
8. Lattice-ordered vector groups
9. Complete lattice-ordered groups
10. Embedding in complete lattice-ordered groups
11. The Cantor extension
12. Ideal systems

SECOND PART: PARTIALLY ORDERED RINGS AND FIELDS

CHAPTER VI. Preliminaries on partially ordered rings
1. Partial order on rings and fields
2. Examples
3. Ordering of rings of quotients
4. Embedding in a ring with identity

CHAPTER VII. Extensions of partial orders in rings
1. Extension to a full order; O-rings
2. O-rings without divisors of zero
3. Real closed commutative fields
4. Intersection of full orders
5. Vector rings

CHAPTER VIII. Fully ordered rings and fields
1. Archimedean fully ordered rings
2. The Archimedean classes
3. O-rings with divisors of zero
4. o-simple fully ordered rings
5. Formal power series fields
6. Completion of fully ordered fields

CHAPTER IX. Lattice-ordered rings
1. General properties of lattice-ordered rings
2. Function rings
3. The $L$-radical of lattice-ordered rings

THIRD PART: PARTIALLY ORDERED SEMIGROUPS

CHAPTER X. Partial orders on semigroups
1. Partially ordered groupoids and semigroups
2. Examples
3. The positive and negative cones
4. Semigroups of quotients

CHAPTER XI. Fully ordered semigroups
1. Definitions and preliminary lemmas
2. Archimedean, naturally fully ordered semigroups
3. Subsemigroups of the group of real numbers
4. Archimedean semigroups with anomalous pairs
5. Archimedean classes
6. Ordinal sums
7. Completion of fully ordered semigroups
8. On a class of fully ordered groupoids

CHAPTER XII. Lattice-ordered semigroups
1. Residuals
2. Lattice-ordered semigroups 3. The equivalence of Artin
4. Elements with special properties
5. Unicity statements on meet decompositions
6. Meet decompositions of elements

2.3. Review by: David Sachs,
Pi Mu Epsilon Journal 4 (2) (1965), 80.

This book is a survey of the theory of partially ordered groups, rings, fields and semi groups, and much attention is given to the fully and lattice-ordered structures. The book is reasonably self-contained and contains an extensive bibliography of the articles and books written in this area. The author considers the non-abelian case as well as the abelian situation, and many things are done in great generality. The book is written in a reasonable style, but the author supposes that the reader has a good knowledge of abstract algebra. No exercises are included, but there is a long list of unsolved problems for the research-minded individual. I would recommend this book to any advanced graduate student who is interested in learning something about the algebraic aspects of partially ordered systems. It is an important addition to the mathematical literature.

2.4. Review by: Paul F Conrad.
Mathematical Reviews MR0171864 (30 #2090).

This little book is the first systematic survey of the theory of partially ordered groups, rings and semigroups. The stated purpose is "to present the most essential results to those who want to become acquainted with the subject", and from this point of view the book is an unqualified success. Future authors and referees will have to be familiar with this book and its large working bibliography, and thus avoid the rather tedious duplication of results that have occurred in the past few years. The book is beautifully written; some of the proofs are lifted directly from the literature, but many proofs are entirely new or simplifications of the original proofs.
...
To sum up, this is an excellent book that gives the reader a clear picture of some of the important algebraic results in the theory of partially ordered algebraic systems, and, together with the bibliography, gives a good survey of this branch of mathematics.

3.1. Review by: G Bruns.
Canadian Mathematical Bulletin 14 (1) (1971), 142.

These lecture notes are based on four earlier research papers of the author.

3.2. Review by: S J Bernau.
Mathematical Reviews MR0203436 (34 #3288).

Riesz groups, directed partially ordered groups satisfying the Riesz interpolation property, were studied by the author in 1965. The work under review, based on lectures delivered at Queen's University early in 1966, chiefly contains results for Riesz spaces, vector spaces which are Riesz groups, which follow from the paper cited above and from later papers of the author ...

4.1. From the Preface.

The theory of abelian groups is a branch of algebra which deals with commutative groups. Curiously enough, it is rather independent of general group theory: its basic ideas and methods bear only a slight resemblance to the noncommutative case, and there are reasons to believe that no other condition on groups is more decisive for the group structure than commutativity.

The present book is devoted to the theory of abelian groups. The study of abelian groups may be recommended for two principal reasons: in the first place, because of the beauty of the results which include some of the best examples of what is called algebraic structure theory; in the second place, it is one of the principal motives of new research in module theory (e.g., for every particular theorem on abelian groups one can ask over what rings the same result holds) and there are other areas of mathematics in which extensive use of abelian group theory might be very fruitful (structure of homology groups, etc.).

It was the author's original intention to write a second edition of his book "Abelian Groups" (Budapest, 1958). However, it soon became evident that in the last decade the theory of abelian groups has moved too rapidly for a mere revised edition, and consequently, a completely new book has been written which reflects the new aspects of the theory. Some topics (lattice of subgroups, direct decompositions into subsets, etc.) which were treated in "Abelian Groups" will not be touched upon here.

The twin aims of this book are to introduce graduate students to the theory of abelian groups and to provide a young algebraist with a reasonably comprehensive summary of the material on which research in abelian groups can be based. The treatment is by no means intended to be exhaustive or even to yield a complete record of the present status of the theory - this would have been a Sisyphean task, since the subject has become so extensive and is growing almost from day to day. But the author has tried to be fairly complete in what he considers as the main body of up-to-date abelian group theory, and the reader should get a considerable amount of knowledge of the central ideas, the basic results, and the fundamental methods. To assist the reader in this, numerous exercises accompany the text; some of them are straightforward, others serve as additional theory or contain various complements. The exercises are not used in the text except for other exercises, but the reader is advised to attempt some exercises to get a better under-standing of the theory. No mathematical knowledge is presupposed beyond the rudiments of abstract algebra, set theory, and topology; however, a certain maturity in mathematical reasoning is required.

The selection of material is unavoidably somewhat subjective. The main emphasis is on structural problems, and proper place is given to homological questions and to some topological considerations. A serious attempt has been made to unify methods, to simplify presentation, and to make the treatment as self-contained as possible. The author has tried to avoid making the discussion too abstract or too technical. With this view in mind, some significant results could not be treated here and maximum generality has not been achieved in those places where this would entail a loss of clarity or a lot of technicalities.

Volume I presents what is fundamental in abelian groups together with the homological aspects of the theory, while Volume II is devoted to the structure theory and to applications. Each volume has a Bibliography listing those works on abelian groups which are referred to in the text. The author has tried to give credit wherever it belongs. In some instances, however, especially in the exercises, it was nearly impossible to credit ideas to their original discoverers. At the end of each chapter, some comments are made on the topics of the chapter, and some further results and generalisations (also to modules) are mentioned which a reader may wish to pursue. Also, research problems are listed which the author thought interesting.

4.2. Contents of Infinite abelian groups. Vol. I.

Preface

I. Preliminaries
1. Definitions
2. Maps and Diagrams
3. The Most Important Types of Groups
4. Modules
5. Categories of Abelian Groups
6. Functorial Subgroups and Quotient Groups
7. Topologies in Groups
Notes

II. Direct Sums
8. Direct Sums and Direct Products
9. Direct Summands
10. Pullback and Pushout Diagrams
11. Direct Limits
12. Inverse Limits
13. Completeness and Completions Notes

III. Direct Sums of Cyclic Groups
14. Free Abelian Groups-Defining Relations
15. Finitely Generated Groups
16. Linear Independence and Rank
17. Direct Sums of Cyclic $p$-Groups
18. Subgroups of Direct Sums of Cyclic Groups
19. Countable Free Groups Notes

IV. Divisible Groups
20. Divisibility
21. Injective Groups
22. Systems of Equations
23. The Structure of Divisible Groups
24. The Divisible Hull
25. Finitely Cogenerated Groups Notes

V. Pure Subgroups
26. Purity
27. Bounded Pure Subgroups
28. Quotient Groups Modulo Pure Subgroups
29. Pure-Exact Sequences
30. Pure-Projectivity and Pure-Injectivity
31. Generalizations of Purity
Notes

VI. Basic Subgroups
32. p-Basic Subgroups
33. Basic Subgroups of p-Groups
34. Further Results on p-Basic Subgroups
35. Different p-Basic Subgroups
36. Basic Subgroups Are Endomorphic Images
37. The Ulm Sequence Notes

VII. Algebraically Compact Groups
38. Algebraic Compactness
39. Complete Groups
40. The Structure of Algebraically Compact Groups
41. Pure-Essential Extensions
42. More about Algebraically Compact Groups Notes

VIII. Homomorphism Groups
43. Groups of Homomorphisms
44. Exact Sequences for Hom
45. Certain Subgroups of Hom
46. Homomorphism Groups of Torsion Groups
47. Character Groups
48. Duality between Discrete Torsion and 0-Dimensional Compact Groups Notes

IX. Groups of Extensions
49. Group Extensions
50. Extensions as Short Exact Sequences
51. Exact Sequences for Ext
52. Elementary Properties of Ext
53. The Functor Pext
54. Cotorsion Groups
55. The Structure of Cotorsion Groups
56. The Ulm Factors of Cotorsion Groups
57. Applications to Ext
58. Injective Properties of Cotorsion Groups Notes

X. Tensor and Torsion Products
59. The Tensor Product
60. Exact Sequences for Tensor Products
61. The Structure of Tensor Products
62. The Torsion Product
63. Exact Sequences for Tor
64. The Structure of Torsion Products Notes

4.3. Review by: Charles K Megibben.
American Scientist 59 (1) (1971), 120.

This is the first volume of a completely new monograph on the theory of abelian groups and reflects the many contributions made since the appearance of the author's outstanding Abelian Groups (Budapest, 1958), hereinafter referred to as AG. The new book "presents what is fundamental in abelian groups together with the homological aspects of the theory," structure theory and applications being reserved for the forthcoming second volume. For the most part then, this volume might be said to be devoted to the "soft" portions of the theory.

The first four chapters treat the standard elementary results including direct sums of cyclic groups and divisible groups. The categorical point of view is emphasised, and noteworthy are the discussions of direct and inverse limits and of topological completions. The fifth chapter is devoted to the basic facts concerning pure subgroups and certain recent generalisations of the concept of purity. Pure-injectives (= algebraically compact groups) are introduced here and are returned to for an exhaustive treatment in the seventh chapter. The intervening chapter deals with basic subgroups introduced via the more general notion of a p-basic subgroup. The final three chapters are devoted almost exclusively to the homological aspects of the subject, the titles being Homomorphism Groups, Groups of Extensions, and Tensor and Torsion Products. The chapter on homomorphism groups features the algebraic structure of character groups and of Hom $(A, B)$ for $A$ and $B$ torsion groups and an elementary presentation of the Pontryagin duality between discrete and compact abelian groups. Unlike AG, the treatment of the functor Ext is by means of Baer sums à la Mac Lane. Thus the connecting homomorphism between Hom and Ext arises naturally, the failure to establish this homomorphism being a major weakness of AG. Cotorsion groups and the important subfunctor Pext are also treated here.

The typography is excellent and the book appears to be free of misprints. The references to the periodical literature exhibit that high level of scholarship we have come to expect from Professor Fuchs. There are 653 exercises of varying difficulty and 50 research problems. To each of the ten chapters are appended notes which contain, in addition to historical remarks, many valuable comments concerning generalisations to modules. The quality of exposition is consistently high, though the captious reader will find frequent solecisms attributable no doubt to English not being Professor Fuchs's native tongue.

This book will surely become an indispensable reference to the specialist, as well as an invaluable text for the serious student of the subject. Though it is well within the grasp of most second-year graduate students, its value as an introduction to abelian groups is more doubtful, for after the nearly three hundred pages of the first volume we have not even arrived at Ulm's theorem. This seems inevitable, however. Twelve years ago Professor Fuchs was able to put into the first 250 pages of AG all that one needed to know in order to do research on abelian groups; largely as a result of the influence of AG, this is no longer possible.

4.4. Review by: J Rotman.
Mathematical Reviews MR0255673 (41 #333).

There are only three other books that deal seriously with abelian groups: I Kaplansky's Infinite abelian groups, 1954; , A G Kurosh's The theory of groups, 1953; and the author's Abelian groups, 1958. In the dozen years since the last of these appeared, many important theorems have been discovered, usually by homological techniques. Indeed, the subject might be called "modules over a principal ideal domain" to emphasise that it is a chapter of that part of homological algebra dealing with modules over arbitrary rings. Many group-theoretic results are best understood in this broader context. Of course, the deepest and most interesting theorems about groups are those that do not hold for all $R$-modules.

This book, the first of two volumes, gives the basic definitions, constructions, and first theorems about abelian groups; the deeper results will appear in the second volume. Even though this book is a prelude to the next, a reader could also regard it as an introduction to modules; many general constructions are easier in the special case of groups, and, along the way, there are very interesting examples.

5.1. Review by: E A Walker.
Mathematical Reviews MR0349869 (50 #2362).

Volume I was a prelude to Volume II and has been reviewed ... The two volumes are not just an expanded version of the author's earlier book Abelian groups, 1958. The subject today is vastly different from what it was 17 years ago, and these two volumes give a faithful representation of the subject. (There have been, however, some significant developments since Volume II was written.) The author's stated aims are "to introduce graduate students to the theory of abelian groups and to provide a young algebraist with a reasonably comprehensive summary of the material on which research in abelian groups can be based". These aims are certainly met. The reviewer can muster no real quarrel with the choice of topics. Volumes I and II represent a truly masterful scholarly achievement, and will certainly be both the standard references and standard texts in the subject for years to come.

6.1. Review by: R S Pierce.
Mathematical Reviews MR0569744 (82f:20081).

These lucidly written notes give an exposition of two active areas of research in abelian group theory. The first half of the monograph deals with valued vector spaces, that is, pairs $(V, v$) in which $V$ is a vector space over a field $F$ and $v$ is a mapping from $V$ to a totally ordered class in which every nonempty set has a supremum. The mapping $v$ is assumed to satisfy axioms that are analogues of the conditions that define a logarithmic, non-Archimedean valuation. The second half of the notes deals with recent work on mixed abelian groups that has evolved from Warfield's fundamental discoveries concerning simply presented groups. These topics have a common ancestor: totally projective $p$-groups. The connection between totally projective $p$-groups and valued vector spaces is made by way of Ulm's theorem. Indeed, the height function on the socle of a $p$-group is the premier example of a valued vector space, at least for this audience. The original impetus for studying valued vector spaces was the desire to clarify Hill's proof that the totally projective p-groups are classified by their Ulm invariants. The exposition in this monograph leaves no doubt that this objective has been achieved in style. The material in the second part of the note is related to the theory of totally projective $p$-groups by way of the Crawley-Hales theory of simply presented groups, that is groups that are defined by relations involving at most two generators. On the one hand, Crawley and Hales proved that the simply presented $p$-groups coincide with the totally projective $p$-groups; Warfield went on to show that the mixed simply presented groups also have a rich structure theory. The results that the author surveys come from papers of Warfield, Stanton, and the Las Cruces consortium: Arnold, Hunter, Richman, and the Walkers.

7.1. Review by: Alberto Facchini.
Mathematical Reviews MR0709258 (84k:13009).

These notes are based on a series of lectures given by the author in 1982. As the title suggests, this work treats modules over valuation domains, that is, integral domains whose ideals are totally ordered with respect to inclusion. ... The book is a well-written and useful summary of some recent work in the field of modules over valuation rings; it takes the reader from the basics up to part of the current research frontier. As the author says in the preface, some topics have not been included in the discussion. ... the book is clear, elegant and largely self-contained, and can be read and appreciated by a wide audience.

8.1. From the Publisher.

This book initiates a systematic, in-depth study of Modules Over Valuation Domains. It introduces the theory of modules over commutative domains without finiteness conditions and examines frontiers of current research in modules over valuation domains. It represents a unique effort to combine ideas from abelian group theory, in a large scale, with powerful techniques developed in module theory. This volume surveys the background material on valuation rings, modules and homological algebra ... features new results for important classes of modules such as finitely generated, divisible, pure-injective, and projective dimension one - never published before ... contains exercises and research problems - offering guidance for independent and creative study ... and provides historical notes, comments, and an extensive bibliography. Mathematicians and advanced graduate-level mathematics students interested in module theory, abelian group theory, and commutative ring theory can stay abreast of the latest advances with Modules Over Valuation Domains.

8.2. From the Preface.

The theory of modules over commutative non-Noetherian domains and, in particular, over valuation domains goes back thirty years or so. Results by several authors, including I Kaplansky, B Osofsky, R B Warfield, and above all E Matlis, were the leading forces in the development. Our debt to their works is enormous.

This volume has evolved as an outgrowth of our search for a systematic treatment of modules over valuation domains. Important results (by other authors) which are not intimately related to ours will not be discussed here. On the other hand, some results which are so far quite meagre will be included wherever it seems worth-while at least to set out the problems. In fact, there are many exciting problems in this subject. In our presentation, we rely heavily both on material available in the literature and on our own research, including joint papers with other authors and works of doctoral students. As we prefer the general framework of arbitrary domains, wherever not too much extra apparatus is required, we state the results in a general form, though it has not been our endeavour to phrase the theorems under most general hypotheses. We are fully aware of the fact that a full-fledged theory over arbitrary domains would be a formidable task, beyond the scope of the present volume and certainly far beyond the authors' knowledge.

Numerous papers on modules have been inspired by abelian group theory which has been a continuous source of ideas for new research. Our treatment owes much more to abelian group theory than is apparent from the text. Throughout our work, enormous stimulus has been given by ideas and methods from abelian groups. A reader familiar with abelian groups will frequently discover their catalytic effects on our subject, even at places which have apparently nothing to do with them. Though our choice of topics occasionally reflects a close kinship to abelian groups, we have not compromised with the guiding principle that a viable theory ought to be developed on its own foundation and to satisfy its own needs.

We have tried to write this volume with both experts and students in mind. We start from no more than a reasonable acquaintance with elementary facts concerning commutative rings, modules and homological algebra. Chapter I collects most of the background material on valuation rings, Chapter II on modules and Chapter III on homological algebra. The remaining chapters can be read, in general, consecutively, without referring to the rest of the volume, though occasionally it was unavoidable not to invoke results developed in later chapters. The chapters end with comments and open problems.
...
The authors have given series of talks on the subject of this volume at several universities, including Tulane University, Università di Padova, Udine, Universities of Calgary, Florida, Orange Free State and Pretoria, Universität Essen, Bar Ilan University and Charles University of Prague. We wish to extend our gratitude to these universities.

8.3. Contents of Modules over valuation domains.

Preface

I. VALUATION RINGS
1. Valuation rings
2. Totally ordered abelian groups
3. Valuations
4. Ideals
5. Maximal and almost maximal valuation rings
6. Prüfer domains
Notes (Problem 1)

II. PRELIMINARIES ON MODULES
1. Modules
2. Divisibility
3. Relative divisibility (RD)
4. Pure submodules
5. Lemmas on pure submodules
6. Cyclic purity
7. Modules with local endomorphism rings
Notes

III. HOMOLOGICAL PRELIMINARIES
1. Homological background
2. Lemmas on Hom and Ext
3. Lemmas on tensor and torsion products

IV. PROJECTIVITY AND PROJECTIVE DIMENSION
1. Projective and flat modules
2. Projective dimension
3. Projective dimensions of torsion-free modules
4. Projective dimension one
5. Tight systems
6. Quasi-projectivity
Notes (Problems 2, 3)

V. TOPOLOGY AND FILTRATIONS
1. The R-topology
2. R-complete modules
3. Filtration and ultracompleteness
4. The annihilator filtration
5. R-ultracomplete modules
Notes (Problem 4)

VI. DIVISIBILITY AND INJECTIVITY
1. Divisible modules
2. h-divisible modules
3. Divisible modules of projective dimension one
4. Injective modules
5. The injective dimension
6. Quasi-injectivity
Notes (Problems 5, 6)

VII. UNISERIAL MODULES
1. Uniserial modules
2. Endomorphism rings of uniserial modules
3. Non-standard uniserial modules
4. Direct sums of uniserial modules
Notes (Problem 7)

VIII. HEIGHTS AND INDICATORS
1. Heights
2. Equiheight submodules
3. Indicators
4. Irregularities of indicators
5. Smoothness
Notes (Problem 8)

IX. FINITELY GENERATED AND POLYSERIAL MODULES
1. Finitely generated modules
2. The Goldie dimension
3. Indecomposable finitely generated modules
4. Decompositions of finitely generated modules
5. Polyserial modules
Notes (Problems 9-17)

Χ. INVARIANTS AND BASIC SUBMODULES
1. α-Invariants
2. α-Invariants of equiheight submodules
3. α-Basic submodules
4. Modules with trivial α-invariants
Notes (Problems 18, 19)

XI. RD-INJECTIVITY AND PURE-INJECTIVITY
1. RD-injective modules
2. Pure-injective modules
3. Pure-injective modules over Prüfer domains
4. Pure-injectivity over valuation domains
5. Pure-injective hulls of polyserial modules
Notes (Problems 20-23)

XII. TORSION-COMPLETE AND COTORSION MODULES
1. Torsion-complete modules
2. Torsion-ultracomplete modules
3. Cotorsion modules
4. The cotorsion hull
Notes (Problem 24)

XIII. TORSION MODULES
1. Embedding in pure polyserial submodules
2. Separable modules
3. Submodules of separable modules
4. Direct sums of cyclic modules
5. Torsion modules of projective dimension one
6. Modules with zero $a$-invariants
Notes (Problem 25)

XIV. TORSION-FREE MODULES
1. Preliminaries
2. Completely decomposable modules
3. Finite rank modules over almost maximal valuation domains
4. Rank one dense basic submodules
5. Chains of pure submodules
6. Pure submodules of free modules
7. Slender modules
Notes (Problem 26)

8.4. Review by: Alberto Facchini.
Mathematical Reviews MR0786121 (86h:13008).

This book is devoted to modules over valuation domains, that is, integral domains whose ideals are totally ordered under inclusion. It contains a wealth of material, most of it appearing in book form for the first time. It also contains some results never before published. We summarise from the authors' introduction: "The theory of modules over commutative non-Noetherian domains, and in particular over valuation domains, goes back thirty years or so. Results by several authors, including I Kaplansky, B Osofsky, R B Warfield and above all E Matlis were the leading forces in the development. This volume has evolved as an outgrowth of our search for a systematic treatment of modules over valuation domains. Some results (by other authors) not intimately related to ours will not be discussed here. There are many exciting problems in this subject. In our presentation we rely heavily both on material available in the literature and on our own research, including joint papers with other authors and works of doctoral students. As we prefer the general framework of arbitrary domains, wherever not too much extra apparatus is required, we state the results in a general form, though it has not been our endeavour to phrase the theorems under the most general hypotheses."

8.5. Review by: A Buium.
Bulletin mathématique de la Société des Sciences Mathématiques de la République Socialiste de Roumanie, Nouvelle Série 32 (80) (2) (1988), 186.

The book under review is a research monograph devoted to the study of the category of modules over a valuation domain. The theory is developed having in mind both the analogy with the theory of abelian groups and the possibility of further generalisation to modules over arbitrary domains. Many techniques from the abelian group theory can be extended to deal with modules over valuation domains. But a series of new phenomena occur too, leading to new relevant concepts (polyserial, heights, RD-injectivity, a. s. o.).

The book is interesting for those working in commutative algebra but addresses the non-expert readers as well. The field is widely open for further research.

8.6. Review by: Tiberiu Dumitrescu.
Bulletin mathématique de la Société des Sciences Mathématiques de la RépubliqueSocialiste de Roumanie Nouvelle Série 31 (79) (2) (1987), 181-182.

In this book the authors develop a structure theory of modules over valuation domains (i.e. domains whose ideals form a chain under inclusion) adapting many of the methods of the abelian group theory to this more general situation, and also using strong results from the theory of modules over non-Noetherian rings. From the authors' introduction: "It is no secret that the classification of abelian groups had made remarkable advances in the past quarter of a century and reached a stage of logical clarity and exceptional effectiveness. It is therefore reasonable to expect that the theory of abelian groups can provide us with models in order to venture in the as yet unexplored areas of module structures ... We have tried to write this volume with both experts and students in mind. We start from no more than a reasonable acquaintance with elementary facts concerning commutative rings, modules and homological algebra."
...
The exposition is clear and exciting. The book contains a lot of information and is largely self contained.

9. Modules over non-Noetherian domains (2001), by László Fuchs and Luigi Salce.

9.1. From the Publisher.

In this book, the authors present both traditional and modern discoveries in the subject area, concentrating on advanced aspects of the topic. Existing material is studied in detail, including finitely generated modules, projective and injective modules, and the theory of torsion and torsion-free modules. Some topics are treated from a new point of view. Also included are areas not found in current texts, for example, pure-injectivity, divisible modules, uniserial modules, etc. Special emphasis is given to results that are valid over arbitrary domains. The authors concentrate on modules over valuation and Prüfer domains, but also discuss Krull and Matlis domains, h-local, reflexive, and coherent domains. The volume can serve as a standard reference book for specialists working in the area and also is a suitable text for advanced-graduate algebra courses and seminars.

9.2. From the Preface.

Contemporary research in module theory over commutative rings is heavily concentrated on those modules for which either the underlying ring or the modules themselves or both are subject to various finiteness restrictions. This is due not only to the widespread applications in other areas but also to the techniques avail-able. However, a considerable amount of work has been done recently on modules without assuming any finiteness condition, resulting in a rapid development of the subject. This volume deals with the theory of modules over commutative integral domains, paying only scant attention to the structure of the underlying domains and practically ignoring the noetherian case.

In the study of modules, a dramatic change occurs when one abandons finiteness conditions. Although most pleasant features are undoubtedly lost, some nice features still remain. These have served as starting points for new discoveries. Substantial generalisations of classical results required totally new methods, and the development of powerful techniques breathed new life into the theory. As a result, module theory over non-noetherian rings, and, in particular, over non-noetherian domains, became a lively branch of algebra.

We feel that the steppingstone to a study of modules over general domains is the module theory of valuation domains these are perhaps the simplest non-noetherian domains that are not too close to Dedekind domains and their global versions, the Prüfer domains. One could attain substantial understanding of module properties by a careful examination of these two special cases. Until recently, not much was known about the theory of modules over these domains, but the past two decades have seen remarkable developments. It has become increasingly clear that they provide a meeting ground for several branches of algebra and supply a wealth of challenging problems. However, the discussions here will not be confined to modules over these domains: whenever feasible, we pursue our treatment initially without assuming any extra condition on the domains; additional conditions will be introduced only when necessary.

In this volume we have tried to present the bulk of the traditional material and to incorporate recent discoveries on our subject by pulling together the main strands of the theory. However, because of the vastness of the topic, limitations had to be imposed on both the choice of the material and the method of presentation. The theory is replete with aesthetically pleasing and powerful results, but we could not (and we did not even intend to) cover certain basic topics such as a study of direct decompositions, endomorphism rings and automorphism groups, K-theory, modules over specific types of domains, etc. We could just briefly touch upon subjects like generalisations of projectivity and injectivity, and the topological aspects of module theory. No attempt has been made to be exhaustive even in the topics covered; we aimed rather to draw together in a systematic manner the different trends of developments, forging them into a more coherent theory. Our intent was to concentrate on the backbone of the theory and to focus attention on results of theoretical interest. We have deliberately omitted standard material covered in textbooks and monographs and skipped details of proofs of results that can be found in several textbooks on algebra or, in particular, on ring theory.

Needless to say, the focus of our presentation is very personal, reflecting our own interests and research; we did not include areas which are not too close to our research interests. As a result, several important aspects of module theory (even within our self-imposed limitation) are bound to be neglected, and one could argue that various additional topics should have been included in this volume, especially those on which there is no ample survey in the literature. Of course, there is always room for argument as to what topics are more relevant or significant. We believe that we have presented an attractive though by no means exhaustive theory of modules over non-noetherian domains which could prepare the groundwork for a more penetrating assault on the subject and which will hopefully inspire further work in the area. Numerous open problems which the authors thought interesting are listed at the end of the chapters.
***
We mathematicians often endeavour to extend theorems in order to cover a wider area or to get a better insight. In this respect, the theory of abelian groups has been a constant source of inspiration for our work. It is apparent that our treatment owes a great deal to abelian groups: old techniques find new roles, and a number of classical results lend themselves to generalisations.

Additional impetus for our work comes from Dedekind domains. It has been observed that if we focus our attention on modules of projective dimension 1, then some of the useful features of modules over Dedekind domains can be retained. A careful reader will find numerous results in this volume in support of our claim that this class of modules deserves special attention.

The powerful apparatus built up in the noetherian case could hardly be utilised in our general setting. It is unreasonable to expect that the same notions would be of comparable relevance in the general case, so suitable substitutes were pursued. For instance, the fruitful idea of introducing an operation between the studied objects (à la Picard group) led us to the consideration of two groups: the archimedean group $Arch R$ for valuation domains $R$ and the group $Gp R$ of 'clones' attached to any domain $R$. Actually, we went a step further and initiated the systematic study of several emerging Clifford semigroups: the inclusion of Clifford semigroups in our arsenal will allow a more global picture of the subjects.

We have also borrowed ideas from our previous volume [FS] that grew out of our attempt to systematically transplant ideas and methods from abelian group theory to the theory of modules. Since its publication (15 years ago) new methods have been developed which have not only improved the original results, but in fact have extended the theory to a wider class of modules. Whenever it was feasible, we adapted methods which provided a little more mileage than the conventional approach. As was pointed out earlier, our general intention was to treat the problems in full generality and to specialise to individual domains whenever it became inevitable. Notable exceptions were the cases when nothing substantial was available in the general case or when the problems were uninteresting in a more general setup.

9.3. Table of Contents of Modules over non-Noetherian domains.

Preface

List of Symbols

Chapter I. Commutative Domains and Their Modules
1. Generalities on domains
2. Fractional ideals
3. Integral dependence
4. Module categories
5. Lemmas on Hom and Ext
6. Lemmas on tensor and torsion products
7. Divisibility and relative divisibility
8. Pure submodules
9. The exchange property
10. Semilocal endomorphism rings
Notes

Chapter II. Valuation Domains
1. Fundamental properties of valuation domains
2. Totally ordered abelian groups
3. Valuations
4. Ideals of valuation domains
5. The class semigroup
6. Maximal and almost maximal valuation domains
7. Henselian valuation rings
8. Strongly discrete valuation domains
Notes

Chapter III. Examiner Domains
1. Fundamental properties and characterizations
2. Prüfer domains of finite character
3. The class semigroup
4. Lattice-ordered abelian groups
5. Bézout domains
6. Elementary divisor domains
7. Strongly discrete examiner domains
Notes

Chapter IV. More Non-Noetherian Domains
1. Krull domains
2. Coherent domains
3. $h$-Local domains
4. Matlis domains
5. Reflexive domains
Notes

Chapter V. Finitely Generated Modules 1. Cyclic modules
2. Finitely generated modules
3. Finitely presented modules
4. Finite presentations
5. Finitely generated modules over valuation domains
6. Indecomposable finitely generated modules
7. Finitely generated modules with local endomorphism rings
8. Decompositions of finitely generated modules
9. Finitely generated modules without the Krull-Schmidt property
10. Domains whose finitely generated modules are direct sums of cyclics
Notes

Chapter VI. Projectivity and Projective Dimension
1. Projective modules
2. Projective dimension
3. Projective dimension over valuation domains
4. Global projective dimension of Prüfer domains
5. Tight submodules
6. Modules of projective dimension one
7. Equivalent presentations
8. Stacked bases over $h$-local Prüfer domains
9. Flat modules
10. Weak dimension
11. Quasi-projective modules
12. Pure- and RD-projectivity
Notes

Chapter VII. Divisible Modules
1. Divisible modules
2. h-Divisible modules, Matlis domains
3. Divisible modules over valuation domains
4. Categories of divisible modules
5. Indecomposable divisible modules
6. Superdecomposable divisible modules
Notes

Chapter VIII. Topology and Filtration
1. The $R$-topology
2. Complete torsion-free modules. The Matlis category equivalence
3. Completions of ideals
4. $R$-Completions over Matlis domains
5. Cokernels of $R$-completions
6. Weakly cotorsion modules
7. Linear compactness
8. Filtration and ultracompleteness
Notes

Chapter IX. Injective Modules
1. Injectivity
2. Indecomposable injectives
3. Absolute purity
4. Injectives over valuation and Prüfer domains
5. $E$-Injectives
6. Injectives over Krull domains
7. Injective dimension
8. Quasi-injective modules
Notes

Chapter X. Uniserial Modules
1. Generalities on uniserial modules
2. Endomorphism rings of uniserial modules
3. Uniserial modules over valuation domains
4. Existence of non-standard uniserial modules
5. More on the existence of non-standard uniserial modules
6. Kaplansky's problem
7. The threshold submodules
8. Life-span of uniserial modules
9. Uniserial modules of the same level
10. The monoid $Unis R$
Notes

Chapter XI. Heights, Invariants and Basic Submodules
1. Heights
2. Equiheight, nice and balanced submodules
3. Indicators
4. Invariants
5. Basic submodules
6. Modules with trivial invariants
Notes

Chapter XII. Polyserial Modules
1. Polyserial and weakly polyserial modules
2. Direct sums of uniserial modules
3. Monoserial modules
4. Episerial modules and their submodule
5. Direct decompositions of weakly polyserial modules
Notes

Chapter XIII. RD- and Pure-Injectivity
1. RD-Injective modules
2. Pure-injective modules
3. Algebraic compactness
4. Pure-injective modules over Prüfer domains
5. Pure-injective modules over valuation domains
6. Pure-injectivity over coherent domains
7. $\aleph _{1}$-Compact modules
8. Cotorsion modules
Notes

Chapter XIV. Torsion Modules