Quasi-Frobenius Rings

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.

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