THE LIFTING CONDITION AND FULLY INVARIANT SUBMODULES

Abstract

A module M is lifting if for every submodule A of M, there exists a direct summand B of M such that B ≤ A and A/B small in M/B. Every non-cosingular lifting module has the summand sum property. We call any module M FI-lifting if for every fully invariant submodule A of M there exists a direct summand B of M such that B ≤ A and A/B is small in M/B. In contrast to lifting modules, any finite direct sum of FI-lifting modules is FI-lifting.