Injectivity of Quasi-projective Modules, Projectivity of Quasi-injective Modules, and Projective Cover of Injective Modules

Abstract

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 kth upper (resp. lower) Loewy factor of an injective right R-module E and the kth lower (resp. upper) Loewy factor of fR whenever Rf 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 1st upper Loewy factor of E and the 1st lower Loewy factor of Rf.