A ring R is called quasi-Frobenius if it is right or left self-injective and right or left artinian (all four combinations being equivalent). The study of these rings grew out of the theory of representations of a finite group as a group of matrices over a field – the corresponding group algebra is quasi-Frobenius. At the turn of the twentieth century G. Frobenius carried out fundamental work on representations of “hypercomplex systems” – finite dimensional algebras in modern terminology. This topic was revied in the late 1930s and early 1940s by Brauer, Nesbitt, Nakayama, and others in their study of “Frobenius algebras.”. Nakayama introduced quasi-Frobenius rings in 1939 and, in 1951 Ikeda characterized them as the left and right self-injective, left and right artinian rings. The subject is intimately related to duality, the duality from right to left modules induced by the hom functor, and, more importantly for us, the duality related to annihilators. The present extent of the theory is vast, and we make no attempt to be encyclopedic here. Instead we provide an elementary, self-contained account of the basic facts about these rings at a level allowing researchers and graduate students to gain entry to the field. This pays off by giving new insights into some of the outstanding open questions about quasi-Frobenius rings.

Our approach begins by extending many earlier results to a much wider class of rings than heretofore investigated. We call these rings mininjective.