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**Quasi-Frobenius rings.**
*(English)*
Zbl 1042.16009

Cambridge Tracts in Mathematics 158. Cambridge: Cambridge University Press (ISBN 0-521-81593-2/pbk). xvii, 307 p. (2003).

This book is aimed at giving an elementary and self-contained account of the basic facts about quasi-Frobenius (henceforth QF) rings, with the idea of allowing researchers and graduate students to gain entry to the field. QF-rings arose first in the study of representation theory, in the form of Frobenius algebras, of which a large and important class of examples is provided by the group algebras of finite groups. This notion evolved into that of Frobenius rings and quasi-Frobenius rings, the latter having been introduced by Nakayama in 1939. QF-rings are closely related to duality, both in the sense of dualities induced by annihilation between the lattices of left and right ideals of the ring and in the sense of categorical (Morita) dualities between the categories of finitely generated left and finitely generated right modules, induced by the regular module \(_RR_R\). They are also related to injectivity, as it was early observed by Ikeda, who characterized QF-rings in 1951 as the left and right self-injective left and right Artinian rings (any combination of the one-sided versions of these two conditions also suffices).

The authors approach QF-rings from the point of view of injectivity and its generalizations. For this purpose, they use the concept of “right mininjective rings”, defined as the rings such that every isomorphism between simple right ideals is given by left multiplication. This condition (together with some other generalizations of injectivity) is used in the book to study QF-rings and some of their generalizations but not all of them; for example, the hierarchy of QF-1, QF-2 and QF-3 algebras (or rings) that had been introduced by Thrall is not considered. In fact, the authors focus on three important open problems related to QF-rings which motivate the orientation and the contents of the book. These problems are: The Faith conjecture (every left – or right – perfect right self-injective ring is QF), the FGF conjecture (if every finitely generated right \(R\)-module embeds in a free module – \(R\) is then called right FGF, – then \(R\) is QF), and the Faith-Menal conjecture (if, for every \(n\geq 1\), the full matrix ring \(M_n(R)\) is right Noetherian and has the property that every right ideal is an annihilator, then \(R\) is QF).

The book reviews recent work in these problems. It is organized in 9 chapters and 3 appendices, ending with a list of 21 questions, an up-to-date bibliography containing 236 references, and an index of contents. The flavor of the contents may be suggested by the titles of the chapters, which are as follows. Chapter 1: Background. Chapter 2: Mininjective rings. Chapter 3: Semiperfect mininjective rings. Chapter 4: Min-CS rings. Chapter 5: Principally injective and FP rings. Chapter 6: Simple injective and dual rings. Chapter 7: FGF rings. Chapter 8: Johns rings. Chapter 9: A generic example. Appendix A: Morita equivalence. Appendix B: Perfect, semiperfect, and semiregular rings. Appendix C: The Camps-Dicks Theorem.

Summing up, this book is a research monograph with up-to-date information on QF-rings and provides a self-contained and very readable introduction to the topics related with the three conjectures mentioned above.

The authors approach QF-rings from the point of view of injectivity and its generalizations. For this purpose, they use the concept of “right mininjective rings”, defined as the rings such that every isomorphism between simple right ideals is given by left multiplication. This condition (together with some other generalizations of injectivity) is used in the book to study QF-rings and some of their generalizations but not all of them; for example, the hierarchy of QF-1, QF-2 and QF-3 algebras (or rings) that had been introduced by Thrall is not considered. In fact, the authors focus on three important open problems related to QF-rings which motivate the orientation and the contents of the book. These problems are: The Faith conjecture (every left – or right – perfect right self-injective ring is QF), the FGF conjecture (if every finitely generated right \(R\)-module embeds in a free module – \(R\) is then called right FGF, – then \(R\) is QF), and the Faith-Menal conjecture (if, for every \(n\geq 1\), the full matrix ring \(M_n(R)\) is right Noetherian and has the property that every right ideal is an annihilator, then \(R\) is QF).

The book reviews recent work in these problems. It is organized in 9 chapters and 3 appendices, ending with a list of 21 questions, an up-to-date bibliography containing 236 references, and an index of contents. The flavor of the contents may be suggested by the titles of the chapters, which are as follows. Chapter 1: Background. Chapter 2: Mininjective rings. Chapter 3: Semiperfect mininjective rings. Chapter 4: Min-CS rings. Chapter 5: Principally injective and FP rings. Chapter 6: Simple injective and dual rings. Chapter 7: FGF rings. Chapter 8: Johns rings. Chapter 9: A generic example. Appendix A: Morita equivalence. Appendix B: Perfect, semiperfect, and semiregular rings. Appendix C: The Camps-Dicks Theorem.

Summing up, this book is a research monograph with up-to-date information on QF-rings and provides a self-contained and very readable introduction to the topics related with the three conjectures mentioned above.

### MSC:

16L60 | Quasi-Frobenius rings |

16D50 | Injective modules, self-injective associative rings |

16D90 | Module categories in associative algebras |

16L30 | Noncommutative local and semilocal rings, perfect rings |

16-02 | Research exposition (monographs, survey articles) pertaining to associative rings and algebras |