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 $f: M/X_1\to M/X_2$ lifts to an endomorphism $g$ of $M$. A module $M$ is said to be $\mathrm{d}$-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 $\mathrm{d}$-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.