Cogroups and co-rings in categories of associative rings. (English) Zbl 0857.16001

Bergman, George M.; Hausknecht, Adam O.

Mathematical Surveys and Monographs. 45. Providence, RI: American Mathematical Society (AMS). ix, 388 p. (1996).

This book constitutes a hefty contribution to the study of coalgebras and representable functors on well-known varieties of algebras. A broad background is required for its appreciation since it draws on a wide spectrum of algebra, from semigroups and groups through non-commutative rings and bimodules to general category theory.
The bones on which the meat hangs are best described by listing the chapter contents. Chapter 1 is introductory with an overview of the results; Chapter 2 gives a review of coalgebras and representable functors; Chapter 3 deals with representable functors from rings to abelian groups; Chapter 4 contains some digressions on various categories of semigroups; Chapter 5 deals with representable functors from algebras over a field to rings; Chapter 6 deals with representable functors from \(k\)-rings to rings; Chapter 7 deals with representable functors from rings to general groups and semigroups; Chapter 8 deals with representable functors on categories of commutative associative algebras; Chapter 9 deals with representable functors on categories of Lie algebras; Chapter 10 deals with multilinear algebra of representable functors on \(k\)-\(\mathbf{ring}^1\); and the final Chapter 11 discusses directions for further investigation. There is an excellent bibliography of 204 items and a very useful index of symbols.

Reviewer: T.S.Blyth (St.Andrews)

MSC:

16-02	Research exposition (monographs, survey articles) pertaining to associative rings and algebras
08C05	Categories of algebras
16D90	Module categories in associative algebras
18A40	Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
16S10	Associative rings determined by universal properties (free algebras, coproducts, adjunction of inverses, etc.)
16R10	\(T\)-ideals, identities, varieties of associative rings and algebras
17A01	General theory of nonassociative rings and algebras
18D35	Structured objects in a category (MSC2010)
20J15	Category of groups
20M50	Connections of semigroups with homological algebra and category theory
16W30	Hopf algebras (associative rings and algebras) (MSC2000)
17B55	Homological methods in Lie (super)algebras

Keywords:

coalgebras; representable functors; varieties of algebras; categories of semigroups; categories of Lie algebras