SOME PROPERTIES OF EF-EXTENDING RINGS

In [16], Thuyet and Wisbauer considered the extending property for the class of (essentially) finitely generated submodules. A module M is called ef-extending if every closed submodule which contains essentially a finitely generated submodule is a direct summand of M. A ring R is called right ef-extending if R_R is an ef-extending module. We show that a ring R is right ef-extending and the R-dual of every simple left R-module is simple if and only if R is semiperfect right continuous with Sl = Sl ≤^e R_R. We also prove that a ring R is a QF-ring if and only if R is left Kasch and R_R^(ω) is ef-extending if and only if R is right AGP-injective satisfying DCC on right (or left) annihilators and (R ⊕ R)_R is ef-extending.