DIRECT SUMS OF LIFTING MODULES

Abstract

This paper is concerned with when a direct sum of lifting modules is lifting. For example, it is proved that for any ring R, the direct sum M = ⊕i∈IMi is lifting if and only if M is amply supplemented and there exists i ∈ I such that every coclosed submodule K of M with M = K+Mi or M = K + M(I −i) is a direct summand of M. In addition, we prove that for any right perfect ring R, the rightR-module M = M1 ⊕ M2 is lifting if M1 is a lifting right R-module and M2 is a semisimple right R-module such that M2 is N-projective for every proper submodule N of M1.