On dual automorphism-invariant and superfluous ADS-modules

Abstract

Let M and N be two right R-modules. The module M is called dual automorphism-N-invariant if whenever K-1 is a small submodule of M and K-2 is a small submodule of N, then any epimorphism p : M/K-1 -> N/K-2 with small kernel lifts to a homomorphism phi : M -> N. Let pi(1) : P-1 -> M and pi(2) : P-2 -> N be projective covers. M is called an s-ADS-module if for every decomposition M = S circle plus T of M and every supplement T' of S with T' + T = M and (T boolean AND T') << M, we have M = S circle plus T'. It is shown that an amply supplemented R-module M is s-ADS if and only if for each decomposition M = A circle plus B, A and B are relatively dual automorphism invariant.