The author studies the following two conditions for a ring \(R\): (i) every injective \(R\)-module is a lifting module. (ii) every projective \(R\)-module is an extending module. These relations are closely related to the following conditions due to Harada. (*) Every non-small \(R\)-module contains a non-zero injective submodule. (**) Every non-cosmall \(R\)-module contains a non-zero projective direct summand.
The author shows that a given ring \(R\) satisfies (i) iff \(R\) is a right Artinian ring with (*). Also a given ring \(R\) satisfies (ii) iff \(R\) is a ring with acc on right annihilator ideals and \(R\) satisfies (**). Such rings are called right H-rings and right-co-H-rings for the conditions (i) and (ii) resp. Similarly one obtains left H-rings and left co-H-rings. Rings which are both left and right H-rings are called H-rings and similarly for co-H-rings. Examples of H-rings and co-H-rings are generalized uniserial rings. It is shown that if \(R\) is an algebra over a field of finite dimension, then \(R\) is a right H-ring iff \(R\) is a left co-H-ring.