In [K. R. Fuller, on indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115-135, Theorem 3.1] K. R. Fuller gave necessary and sufficient conditions for projective left modules to be injective over a left artinian ring. In [Y. Baba and K. Oshiro, On a theorem of Fuller, preprint] we studied this theorem. In the present paper first we generalize the theorem by giving necessary and sufficient conditions for quasi-projective modules to be injective and ones for quasi-injective modules to be projective. In [K. R. Fuller, On indecomposable injectives over artinian rings, Pacific J. Math. 29 (1969), 115-135, Theorem 2.4] Fuller also gave a natural one to one correspondence between the homogeneous components of the (k) th upper (resp. lower) Loewy factor of an injective right (R)-module (E) and the (k) th lower (resp. upper) Loewy factor of (f R) whenever (R f) is the projective cover of the socle of (E), where (f) is a primitive idempotent in a right artinian ring R. Second we shall give a complete correspondence between simple submodules of the Ist upper Loewy factor of (E) and the 1st lower Loewy factor of (R f). 1993 Academic Press. Inc.