$QF-3'$ rings and Morita duality

In [2] we proved that a one-sided artinianringisQF-3 ifand onlyifitsdouble dual functors preserve monomorphisms. Here with the aidof [3]we prove that the double dual functor over an arbitrary ring preserves monomorphisms of left modules if and only ifitis a leftQF-3' ring. In view of thistheorem resultsin [3]and [4]provide an analogue for QF-3' rings of the Morita-Tachikawa representation theorem forQF-3 rings ([9],Chapter 5). Also we apply it to obtain a characterizationof Morita dualitybetween Grothendieck categoriesthat serves to generalizeOnodera's theorem [7]that cogenerator rings areselfinjective,by showing thatinjectivityis redundant in theclassicalbimodule characterizationof Morita dualityfor categoriesof modules. We denote both the dual functors Homfi(_, RR) and Hom#(_, RR) by ( )*. Recall that there is a natural transformation a: Ir-moo,―( )**, defined via the usual evaluation maps aM: M―>M**. An i?-module M is calledR-reflexive (Rtorsionless)in case aM is an isomorphism (a monomorphism). Also recall that R isleftQF-3' if theinjectiveenvelope E(RR) of rR isi?-torsionless.