On lifting and extending properties on direct sums of hollow uniform modules

Abstract

A module \(M\) is said to be lifting if, for any submodule \(N\) of \(M\), there exists a direct summand \(X\) of \(M\) contained in \(N\) such that \(N/X\) is small in \(M/X\). A module \(M\) is said to satisfy the {\it finite internal exchange property} if, for any direct summand \(X\) of \(M\) and any finite direct sum decomposition \(M = \bigoplus_{i = 1}^n M_i\), there exists a direct summand \(M_i'\) of \(M_i\) \((i = 1, 2, \ldots, n)\) such that \(M = X \oplus (\bigoplus_{i = 1}^n M_i')\). In this paper, we first give characterizations for the square of a hollow and uniform module to be lifting (extending). In addition, we solve negatively the question ``Does any lifting module satisfy the finite internal exchange property?'' as an application of this result.