Quasi-Frobenius algebras provide the basic setting for modular representation theory of finite groups. Indeed, the group algebra of any finite group is quasi-Frobenius. The presented book is a very accessible introduction to basic properties of quasi-Frobenius rings and the modules over them. Rather than dealing with classical representation theory, the authors consider a more general setting of mininjective rings and show that basics of the classical theory can be developed using only elementary module theory in a more general setting. (A ring is right mininjective, if any isomorphism between minimal right ideals is induced by a left multiplication. By a theorem of Ikeda, quasi-Frobenius rings are exactly the right and left mininjective, right and left artinian rings). While basic notions and results on (weak) self-injectivity, CS- and C2-conditions, AB5*, and dualities are developed through Chapters 2-7, the reader is gradually introduced into three challenging open problems: the Faith Conjecture (whether every left or right perfect right self-injective ring is quasi-Frobenius), the FGF-Conjecture (whether the condition that every finitely generated module embeds in a free module implies that the ring is quasi-Frobenius), and the Faith-Menal Conjecture (asking whether any right strongly Johns ring is quasi-Frobenius. A ring R is right Johns, if R is right noetherian and each right ideal of R is an annihilator; R is right strongly Johns, if all full matrix rings Mn(R), n ≥ 1 are Johns). The latter conjecture is investigated in Chapter 8, where an example is given that the conjecture fails if the term ‘strongly’ is omitted. Chapter 9 deals with the Faith Conjecture, providing a generic construction of examples using particular 3x3-upper triangular matrix rings with coefficients in bimodules over division rings. The book concludes with three Appendices: on Morita theory of equivalence, on Bass' theory of perfect rings, and on the Camps-Dicks Theorem (proving that the endomorphism ring of any artinian module is semilocal). The authors have achieved two seemingly incompatible goals: to provide an elementary introduction to the classical theory of quasi-Frobenius rings, and to bring the reader up to the current research in the field. This makes the book interesting both for graduate students and researchers in contemporary module theory.

Reviewer:

jtrl