ON ALMOST N-SIMPLE-PROJECTIVES

The concept of almost N-projectivity is defined in [5] by M. Harada and A. Tozaki to translate the concept "lifting module" in terms of homomorphisms. In [6, Theorem 1] M. Harada defined a little weaker condition "almost N-simple-projecive" and gave the following relationship between them: For a semiperfect ring R and R-modules M and N of finite length, M is almost N-projective if and only if M is almost N-simple-projective. We remove the assumption "of finite length" and give the result in Theorem 5 as follows: For a semiperfect ring R, a finitely generated right R-module M and an indecomposable right R-module N of finite Loewy length, M is almost N-projective if and only if M is almost N-simple-projective. We also see that, for a semiperfect ring R, a finitely generated R-module M and an R-module N of finite Loewy length, M is N-simple-projective if and only if M is N-projective.