ALMOST QF RINGS AND ALMOST QF RINGS

In this paper we assume that every ring R is an associative ring with identity and R is two-sided artinian. The author has defined almost pr ojective modules and almost injective modules in [8], and by making us e of the concept of almost projectives he has defined almost hereditar y rings in [7], whose class contains that of hereditary rings and seri al rings. Similarly to [7] we shall define an almost QF ring, which is a generalization of QF rings. It is well known that an artinian ring R is QF if and only if R is self injective. Following this fact, if R is almost injective as a right R-module, we call R a right almost QF r ing. Analogously we call R a right almost QF(double dagger) ring if ev ery injective is right almost projective. On the other hand, the autho r studied rings with () (resp. (*)*) (see sectional sign 1 for defini tions) in [4]. K. Oshiro called such a ring a right H- (resp. co-H) ri ng in (10]. In this note we shall show that a right almost OF (resp. a lmost QF(double dagger)) ring coincides with a right co-H (resp. H-) r ing. In the final section we shall give a characterization of serial r ings in terms of almost projectives and almost injectives. In the fort hcoming paper [9] we shall study certain conditions under which right almost QF rings are QF or serial.