A Note on Ads* Modules

Abstract

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 circle plus 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 -> S-i/(S-1 boolean AND S-2) with Ker(alpha(i)) = S-j ( i not equal j = 1, 2) can be lifted to an idempotent endomorphism beta(i) of M with beta(i) (M) subset of S-i.