Direct Sums of Hollow-lifting Modules

Abstract

A module M over a ring R is called hollow-lifting if every submodule N of M with M/N hollow contains a direct summand K of M such that N/K is a small submodule of M/K. It is known that a (finite) direct sum of hollow-lifting modules need not be hollow-lifting. In this paper, we show that a quotient of a hollow-lifting module is not hollow-lifting in general, and discuss when a direct sum of hollow-lifting modules is hollow-lifting. We also study relative properties of hollow-lifting modules and investigate direct sums of X-hollow-lifting modules.