A note on ADS* modules

We study the ADS* modules which are the dualizations of ADS modules studied by Alahmadi et al. (J Algebra 352:215–222, 2012). Mainly we prove that an amply supplemented module M is ADS* if and only if M 1 and M 2 are mutually projective whenever ${M = M_{1} \oplus M_{2}}$ if and only if for any direct summand S 1 and a submodule S 2 with M = S 1 + S 2, the epimorphism ${\alpha_{i} : M \longrightarrow S_{i}/(S_{1} \cap S_{2})}$ with Ker (α i ) = S j (i ≠ j = 1, 2) can be lifted to an idempotent endomorphism β i of M with ${\beta_{i}(M) \subseteq S_{i}}$ .