Some generalization of quasi-Frobenius algebras

Introduction. Let 21 be an algebra over a field f. 21 is called quasiFrobenius (QF) if it has a unit element and if every primitive right ideal is dual to a primitive left ideal; or, equivalently, if 21 has a unit element and if every indecomposable direct constituent of the right regular representation is equivalent to an indecomposable direct constituent of the left regular representation. Properties of QF algebras and rings have been treated by Nakayama, Nesbitt, and the author [2, 3, 5, 6](1). Some of the most important properties of QF algebras do not characterize these algebras, but occur in more extensive classes. This leads us to the definitions that follow. Throughout this paper 21 is a f-algebra with unity element. QF-1 Algebra. { is said to be a QF-1 algebra if every faithful representation of 21 is its own second commutator. QF-2 Algebra. 21 is said to be a QF-2 algebra if every primitive left ideal, and every primitive right ideal, of 21 has a unique minimal subideal. A faithful representation Q3 of an algebra 21 is said to be a minimal faithful representation if deletion of any direct constituent of Q3 leaves a nonfaithful representation, that is, if the corresponding space V is the direct sum of VI and V2 with V20 then Q31 is not faithful. QF-3 Algebra. 21 is said to be a QF-3 algebra if it has a unique minimal faithful representation. We shall use the notation QF-12 to describe an algebra which is both QF-1 and QF-2, and so on. Every QF algebra is QF-123. It is the purpose of the present paper to initiate the study of the above classes of algebras. ?1 contains definitions and notations for the paper. ?2 gives an example to show that the class QF-1 is more general than the class QF. ?3 contains a theorem which gives an equivalent definition for QF-2 algebras, ??4 and 5 discuss conditions under which a QF-2 algebra is QF, and give some properties of the Cartan invariants of QF-2 algebras. ?6 contains the proof that every QF-2 algebra is also a QF-3 algebra. ??7 and 8 treat QF-3 algebras and include examples which illustrate the essentially more general character of QF-3 algebras as compared with QF-2 algebras. ?9 treats a necessary condition on the class QF-13. Some of the above definitions are given in the language of ideal theory