Decompositions of Dual Automorphism Invariant Modules over Semiperfect Rings

A module M is called dual automorphism invariant if whenever X 1 and X 2 are small submodules of M , then each epimorphism f : M / X 1 → M / X 2 lifts to an endomorphism g of M . A module M is said to be d-square free (dual square free) if whenever some factor module of M is isomorphic to N 2 for a module N then N = 0. We show that each dual automorphism invariant module over a semiperfect ring which is a small epimorphic image of a projective lifting module is a direct sum of cyclic indecomposable d-square free modules. Moreover, we prove that for each module M over a semiperfect ring which is a small epimorphic image of a projective lifting module (e.g., M is a finitely generated module), M is dual automorphism invariant iff M is pseudoprojective. Also, we give the necessary and sufficient conditions for a dual automorphism invariant module over a right perfect ring to be quasiprojective.