**Mathematical Surveys and Monographs**

Volume: 45;
1996;
388 pp;
Hardcover

MSC: Primary 16; 17; 18;
Secondary 08; 13; 14; 20

**Print ISBN: 978-0-8218-0495-7
Product Code: SURV/45**

List Price: $108.00

AMS Member Price: $86.40

MAA Member Price: $97.20

**Electronic ISBN: 978-1-4704-1276-0
Product Code: SURV/45.E**

List Price: $101.00

AMS Member Price: $80.80

MAA Member Price: $90.90

# Cogroups and Co-rings in Categories of Associative Rings

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*George M. Bergman; Adam O. Hausknecht*

This book studies representable functors among
well-known varieties of algebras. All such functors from associative
rings over a fixed ring \(R\) to each of the categories of
abelian groups, associative rings, Lie rings, and to several others are
determined. Results are also obtained on representable functors on
varieties of groups, semigroups, commutative rings, and Lie
algebras.

The book includes a “Symbol index”, which serves as
a glossary of symbols used and a list of the pages where the topics
so symbolized are treated, and a “Word and phrase index”.
The authors have strived—and succeeded—in creating a
volume that is very user-friendly.

#### Readership

Graduate students and research mathematicians interested in algebra, ring theory, and algebraic geometry.

#### Reviews & Endorsements

Indisputably, Bergman and Hausknecht have done a remarkable job of surveying a broad range of territories … There is no comparable book, or more technically, no book substitutable in significant part for this one. The dedicated reader will soon find herself in love with the very full indexing …

-- Bulletin of the AMS

Constitutes a hefty contribution to the study of coalgebras and representable funtors on well-known varieties of algebras.

-- Zentralblatt MATH

Very elegantly written.

-- Mathematical Reviews

This book is fascinating in many aspects: First, its rigorous, systematic and accessible exposition of the subject makes it a bright landmark at the crossroads of arithmetic and mathematical physics; no doubt it will become a basic reference in random matrix theory. Second, it offers its reader a bouquet of beautiful new results but also leaves the door open to many challenging conjectures. The book will constitute, for those working in the field of \(L\)-functions, a major source of inspiration for the next decades.

-- Mathematical Reviews Featured Review

#### Table of Contents

# Table of Contents

## Cogroups and Co-rings in Categories of Associative Rings

- Contents vii8 free
- Chapter I. Introduction 112 free
- Chapter II. Review of coalgebras and representable functors 920
- 5. Category-theoretic formulations of universal properties, and some other matters 920
- 6. Basic definitions and results of universal algebra 1829
- 7. Some conventions followed throughout this work 1930
- 8. Algebra and coalgebra objects in a category, and representable algebra-valued functors 2031
- 9. Digressions on representable functors 3041

- Chapter III. Representable functors from rings to abelian groups 3546
- Chapter IV. Digressions on semigroups, etc. 6172
- 15. Representable functors to AbBinar[sup(e)] 6172
- 16. Representable functors to abelian semigroups without neutral element – easy results 6374
- 17. Symmetry conditions, and cocommutativity 6879
- 18. Application to AbSemigp-valued functors 7485
- 19. Some observations and questions on rings of symmetric elements 7889
- 20. Representable functors from Semigp[sup(e)] to Semigp[sup(e)] 8293
- 21. Representable functors among varieties of groups and semigroups 89100
- 22. Some related varieties: binars, heaps, and Mal'cev algebras 95106

- Chapter V. Representable functors from algebras over a field to rings 99110
- Chapter VI. Representable functors from k-rings to rings 145156
- Chapter VII. Representable functors from rings to general groups and semigroups 175186
- 33. Functors to Group 175186
- 34. Functors on connected graded rings 179190
- 35. Functors from k-algebras to semigroups: some examples 190201
- 36. Jacobson radicals, and a general construction 194205
- 37. Representable functors to semigroups: toward some conjectures 200211
- 38. Density of invertible elements 202213
- 39. An idempotentless example, and a question on subfunctors 206217

- Chapter VIII. Representable functors on categories of commutative associative algebras 209220
- 40. Some easy examples 210221
- 41. Identities and equational subfunctors 214225
- 42. Idempotents again 224235
- 43. Bialgebras and Hopf algebras 230241
- 44. The Witt vector construction 237248
- 45. The co-ring of integral polynomials 240251
- 46. Generalized integral polynomials 243254
- 47. Representative functions and linearly recursive sequences 246257
- 48. A last tantalizing observation on idempotents 256267

- Chapter IX. Representable functors on categories of Lie algebras 259270
- Chapter X. Multilinear algebra of representable functors on k-Ring[sup(1)] 277288
- Chapter XI. Directions for further investigation 295306
- 57. Other varieties of algebras 295306
- 58. Some miscellaneous remarks 299310
- 59. Prevarieties 303314
- 60. [omitted]-algebras and [omitted]-coalgebras 309320
- 61. Further observations on [omitted] 316327
- 62. Analogs of [omitted] 319330
- 63. Tall-Wraith monads and hermaphroditic functors 326337
- 64. Examples of TW-monads, and further remarks 333344
- 65. The Ehrenfeucht question for semigroups and associative algebras 341352

- References 348359
- Word and phrase index 360371
- Symbol index 380391