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

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.

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 |